LMFDB - Embedded newform 1512.2.bs.a.521.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1512,2,Mod(521,1512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1512.521"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.bs (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.5
Character	$\chi$	$=$	1512.521
Dual form			1512.2.bs.a.1097.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.29668 + 2.24592i) q^{5} +(1.66807 + 2.05367i) q^{7} +(-3.06832 + 1.77150i) q^{11} +(-2.74094 + 1.58248i) q^{13} +(-0.487640 + 0.844616i) q^{17} +(2.11191 - 1.21931i) q^{19} +(-2.92435 - 1.68838i) q^{23} +(-0.862775 - 1.49437i) q^{25} +(-0.267645 - 0.154525i) q^{29} -5.03410i q^{31} +(-6.77533 + 1.08340i) q^{35} +(-3.47324 - 6.01583i) q^{37} +(-6.08175 - 10.5339i) q^{41} +(-5.47630 + 9.48523i) q^{43} +2.86688 q^{47} +(-1.43508 + 6.85132i) q^{49} +(7.81416 + 4.51151i) q^{53} -9.18828i q^{55} +0.439455 q^{59} -3.94033i q^{61} -8.20791i q^{65} +3.65121 q^{67} +5.25055i q^{71} +(-14.0773 - 8.12752i) q^{73} +(-8.75624 - 3.34632i) q^{77} +6.99318 q^{79} +(-7.23851 + 12.5375i) q^{83} +(-1.26463 - 2.19040i) q^{85} +(2.31101 + 4.00279i) q^{89} +(-7.82197 - 2.98928i) q^{91} +6.32426i q^{95} +(-12.4805 - 7.20560i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 12 q^{23} - 24 q^{25} - 18 q^{29} - 6 q^{41} - 6 q^{43} - 36 q^{47} + 6 q^{49} - 12 q^{53} + 36 q^{77} - 12 q^{79} - 18 q^{89} + 6 q^{91}+O(q^{100}) $ 48 * q + 12 * q^23 - 24 * q^25 - 18 * q^29 - 6 * q^41 - 6 * q^43 - 36 * q^47 + 6 * q^49 - 12 * q^53 + 36 * q^77 - 12 * q^79 - 18 * q^89 + 6 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$1$

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
$5$	−1.29668	+	2.24592i	−0.579894	+	1.00441i	0.415596	+	0.909549i	$0.363573\pi$
$5$	−1.29668	+	2.24592i	−0.579894	+	1.00441i	−0.995491	+	0.0948574i	$0.969760\pi$
$6$		0			0
$7$	1.66807	+	2.05367i	0.630471	+	0.776213i
$7$	1.66807	+	2.05367i	0.630471	+	0.776213i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.06832	+	1.77150i	−0.925134	+	0.534126i	−0.885269	−	0.465079i	$-0.846026\pi$
$11$	−3.06832	+	1.77150i	−0.925134	+	0.534126i	−0.0398645	+	0.999205i	$0.512693\pi$
$12$		0			0
$13$	−2.74094	+	1.58248i	−0.760200	+	0.438902i	−0.829367	−	0.558703i	$-0.811299\pi$
$13$	−2.74094	+	1.58248i	−0.760200	+	0.438902i	0.0691677	+	0.997605i	$0.477966\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.487640	+	0.844616i	−0.118270	+	0.204850i	−0.919082	−	0.394066i	$-0.871068\pi$
$17$	−0.487640	+	0.844616i	−0.118270	+	0.204850i	0.800812	+	0.598916i	$0.204401\pi$
$18$		0			0
$19$	2.11191	−	1.21931i	0.484506	−	0.279730i	−0.237786	−	0.971318i	$-0.576422\pi$
$19$	2.11191	−	1.21931i	0.484506	−	0.279730i	0.722293	+	0.691588i	$0.243088\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.92435	−	1.68838i	−0.609770	−	0.352051i	0.163106	−	0.986609i	$-0.447849\pi$
$23$	−2.92435	−	1.68838i	−0.609770	−	0.352051i	−0.772875	+	0.634558i	$0.781182\pi$
$24$		0			0
$25$	−0.862775	−	1.49437i	−0.172555	−	0.298874i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.267645	−	0.154525i	−0.0497004	−	0.0286946i	0.474944	−	0.880016i	$-0.342468\pi$
$29$	−0.267645	−	0.154525i	−0.0497004	−	0.0286946i	−0.524644	+	0.851322i	$0.675802\pi$
$30$		0			0
$31$		−	5.03410i		−	0.904150i	−0.891980	−	0.452075i	$-0.850684\pi$
$31$		−	5.03410i		−	0.904150i	0.891980	−	0.452075i	$-0.149316\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.77533	+	1.08340i	−1.14524	+	0.183128i
$36$		0			0
$37$	−3.47324	−	6.01583i	−0.570997	−	0.988996i	−0.996464	−	0.0840218i	$-0.973223\pi$
$37$	−3.47324	−	6.01583i	−0.570997	−	0.988996i	0.425467	−	0.904974i	$-0.360110\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.08175	−	10.5339i	−0.949810	−	1.64512i	−0.745822	−	0.666146i	$-0.767943\pi$
$41$	−6.08175	−	10.5339i	−0.949810	−	1.64512i	−0.203988	−	0.978973i	$-0.565390\pi$
$42$		0			0
$43$	−5.47630	+	9.48523i	−0.835128	+	1.44648i	0.0587983	+	0.998270i	$0.481273\pi$
$43$	−5.47630	+	9.48523i	−0.835128	+	1.44648i	−0.893926	+	0.448214i	$0.852060\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.86688			0.418177			0.209089	−	0.977897i	$-0.432950\pi$
$47$	2.86688			0.418177			0.209089	+	0.977897i	$0.432950\pi$
$48$		0			0
$49$	−1.43508	+	6.85132i	−0.205012	+	0.978760i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.81416	+	4.51151i	1.07336	+	0.619703i	0.929097	−	0.369836i	$-0.120586\pi$
$53$	7.81416	+	4.51151i	1.07336	+	0.619703i	0.144261	+	0.989540i	$0.453920\pi$
$54$		0			0
$55$		−	9.18828i		−	1.23895i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.439455			0.0572121			0.0286061	−	0.999591i	$-0.490893\pi$
$59$	0.439455			0.0572121			0.0286061	+	0.999591i	$0.490893\pi$
$60$		0			0
$61$		−	3.94033i		−	0.504507i	−0.967661	−	0.252254i	$-0.918828\pi$
$61$		−	3.94033i		−	0.504507i	0.967661	−	0.252254i	$-0.0811717\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	8.20791i		−	1.01807i
$66$		0			0
$67$	3.65121			0.446067			0.223033	−	0.974811i	$-0.428404\pi$
$67$	3.65121			0.446067			0.223033	+	0.974811i	$0.428404\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.25055i			0.623126i	0.950226	+	0.311563i	$0.100852\pi$
$71$			5.25055i			0.623126i	−0.950226	+	0.311563i	$0.899148\pi$
$72$		0			0
$73$	−14.0773	−	8.12752i	−1.64762	−	0.951254i	−0.978014	−	0.208539i	$-0.933129\pi$
$73$	−14.0773	−	8.12752i	−1.64762	−	0.951254i	−0.669607	−	0.742716i	$-0.733537\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.75624	−	3.34632i	−0.997866	−	0.381349i
$78$		0			0
$79$	6.99318			0.786795			0.393397	−	0.919369i	$-0.371300\pi$
$79$	6.99318			0.786795			0.393397	+	0.919369i	$0.371300\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.23851	+	12.5375i	−0.794529	+	1.37616i	0.128609	+	0.991695i	$0.458949\pi$
$83$	−7.23851	+	12.5375i	−0.794529	+	1.37616i	−0.923138	+	0.384469i	$0.874384\pi$
$84$		0			0
$85$	−1.26463	−	2.19040i	−0.137168	−	0.237582i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.31101	+	4.00279i	0.244967	+	0.424295i	0.962122	−	0.272619i	$-0.0878897\pi$
$89$	2.31101	+	4.00279i	0.244967	+	0.424295i	−0.717156	+	0.696913i	$0.754556\pi$
$90$		0			0
$91$	−7.82197	−	2.98928i	−0.819965	−	0.313362i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			6.32426i			0.648855i
$96$		0			0
$97$	−12.4805	−	7.20560i	−1.26720	−	0.731617i	−0.292742	−	0.956192i	$-0.594568\pi$
$97$	−12.4805	−	7.20560i	−1.26720	−	0.731617i	−0.974457	+	0.224574i	$0.927901\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.31460	+	7.47311i	0.429319	+	0.743602i	0.996813	−	0.0797753i	$-0.0254203\pi$
$101$	4.31460	+	7.47311i	0.429319	+	0.743602i	−0.567494	+	0.823378i	$0.692087\pi$
$102$		0			0
$103$	−5.28044	−	3.04866i	−0.520297	−	0.300394i	0.216759	−	0.976225i	$-0.430451\pi$
$103$	−5.28044	−	3.04866i	−0.520297	−	0.300394i	−0.737056	+	0.675832i	$0.763785\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.8841	+	8.01599i	−1.34223	+	0.774935i	−0.987134	−	0.159896i	$-0.948884\pi$
$107$	−13.8841	+	8.01599i	−1.34223	+	0.774935i	−0.355093	+	0.934831i	$0.615551\pi$
$108$		0			0
$109$	2.78659	−	4.82652i	0.266907	−	0.462297i	−0.701154	−	0.713010i	$-0.747332\pi$
$109$	2.78659	−	4.82652i	0.266907	−	0.462297i	0.968062	+	0.250713i	$0.0806649\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.2144	−	8.20669i	1.33718	−	0.772021i	0.350791	−	0.936454i	$-0.385913\pi$
$113$	14.2144	−	8.20669i	1.33718	−	0.772021i	0.986388	+	0.164433i	$0.0525794\pi$
$114$		0			0
$115$	7.58392	−	4.37858i	0.707204	−	0.408305i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.54798	+	0.407431i	−0.233573	+	0.0373492i
$120$		0			0
$121$	0.776398	−	1.34476i	0.0705816	−	0.122251i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.49185			−0.759534
$126$		0			0
$127$	−18.0255			−1.59951			−0.799754	−	0.600328i	$-0.795037\pi$
$127$	−18.0255			−1.59951			−0.799754	+	0.600328i	$0.795037\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.21967	+	9.04074i	−0.456045	+	0.789893i	−0.998748	−	0.0500320i	$-0.984068\pi$
$131$	−5.21967	+	9.04074i	−0.456045	+	0.789893i	0.542703	+	0.839925i	$0.317401\pi$
$132$		0			0
$133$	6.02688	+	2.30326i	0.522597	+	0.199718i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.33412	−	1.92496i	0.284853	−	0.164460i	−0.350765	−	0.936463i	$-0.614078\pi$
$137$	3.33412	−	1.92496i	0.284853	−	0.164460i	0.635619	+	0.772003i	$0.280745\pi$
$138$		0			0
$139$	−18.7634	+	10.8330i	−1.59149	+	0.918847i	−0.598437	+	0.801170i	$0.704211\pi$
$139$	−18.7634	+	10.8330i	−1.59149	+	0.918847i	−0.993052	+	0.117677i	$0.962455\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.60672	−	9.71113i	0.468858	−	0.812085i
$144$		0			0
$145$	0.694102	−	0.400740i	0.0576420	−	0.0332796i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.22975	−	4.75145i	−0.674207	−	0.389254i	0.123462	−	0.992349i	$-0.460600\pi$
$149$	−8.22975	−	4.75145i	−0.674207	−	0.389254i	−0.797669	+	0.603096i	$0.793934\pi$
$150$		0			0
$151$	10.6690	+	18.4793i	0.868232	+	1.50382i	0.863802	+	0.503831i	$0.168077\pi$
$151$	10.6690	+	18.4793i	0.868232	+	1.50382i	0.00442944	+	0.999990i	$0.498590\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	11.3062	+	6.52763i	0.908134	+	0.524312i
$156$		0			0
$157$			2.43102i			0.194016i	0.995284	+	0.0970082i	$0.0309273\pi$
$157$			2.43102i			0.194016i	−0.995284	+	0.0970082i	$0.969073\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.41067	−	8.82197i	−0.111176	−	0.695269i
$162$		0			0
$163$	8.31072	+	14.3946i	0.650946	+	1.12747i	0.982894	+	0.184174i	$0.0589609\pi$
$163$	8.31072	+	14.3946i	0.650946	+	1.12747i	−0.331948	+	0.943298i	$0.607706\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.6037	+	18.3661i	0.820535	+	1.42121i	0.905284	+	0.424806i	$0.139658\pi$
$167$	10.6037	+	18.3661i	0.820535	+	1.42121i	−0.0847489	+	0.996402i	$0.527009\pi$
$168$		0			0
$169$	−1.49150	+	2.58336i	−0.114731	+	0.198720i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.1812			0.774066			0.387033	−	0.922066i	$-0.373500\pi$
$173$	10.1812			0.774066			0.387033	+	0.922066i	$0.373500\pi$
$174$		0			0
$175$	1.62977	−	4.26456i	0.123199	−	0.322371i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.589482	−	0.340338i	−0.0440600	−	0.0254380i	0.477808	−	0.878464i	$-0.341431\pi$
$179$	−0.589482	−	0.340338i	−0.0440600	−	0.0254380i	−0.521868	+	0.853026i	$0.674765\pi$
$180$		0			0
$181$		−	19.1298i		−	1.42191i	−0.703239	−	0.710954i	$-0.748263\pi$
$181$		−	19.1298i		−	1.42191i	0.703239	−	0.710954i	$-0.251737\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	18.0148			1.32447
$186$		0			0
$187$		−	3.45541i		−	0.252684i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	21.1511i		−	1.53044i	−0.643768	−	0.765221i	$-0.722630\pi$
$191$		−	21.1511i		−	1.53044i	0.643768	−	0.765221i	$-0.277370\pi$
$192$		0			0
$193$	12.8284			0.923410			0.461705	−	0.887033i	$-0.347238\pi$
$193$	12.8284			0.923410			0.461705	+	0.887033i	$0.347238\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			22.2854i			1.58777i	0.608069	+	0.793884i	$0.291944\pi$
$197$			22.2854i			1.58777i	−0.608069	+	0.793884i	$0.708056\pi$
$198$		0			0
$199$	6.65637	+	3.84306i	0.471858	+	0.272427i	0.717017	−	0.697056i	$-0.245507\pi$
$199$	6.65637	+	3.84306i	0.471858	+	0.272427i	−0.245159	+	0.969483i	$0.578840\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.129108	−	0.807412i	−0.00906162	−	0.0566692i
$204$		0			0
$205$	31.5444			2.20316
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.32002	+	7.48250i	−0.298822	+	0.517575i
$210$		0			0
$211$	8.33679	+	14.4397i	0.573928	+	0.994073i	0.996157	+	0.0875838i	$0.0279146\pi$
$211$	8.33679	+	14.4397i	0.573928	+	0.994073i	−0.422229	+	0.906489i	$0.638752\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−14.2021	−	24.5987i	−0.968572	−	1.67762i
$216$		0			0
$217$	10.3383	−	8.39722i	0.701813	−	0.570041i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	3.08672i		−	0.207635i
$222$		0			0
$223$	4.45390	+	2.57146i	0.298255	+	0.172198i	0.641659	−	0.766990i	$-0.278247\pi$
$223$	4.45390	+	2.57146i	0.298255	+	0.172198i	−0.343404	+	0.939188i	$0.611580\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.99271	−	10.3797i	−0.397750	−	0.688924i	0.595698	−	0.803209i	$-0.296876\pi$
$227$	−5.99271	−	10.3797i	−0.397750	−	0.688924i	−0.993448	+	0.114285i	$0.963542\pi$
$228$		0			0
$229$	−2.25955	−	1.30455i	−0.149316	−	0.0862074i	0.423481	−	0.905905i	$-0.360808\pi$
$229$	−2.25955	−	1.30455i	−0.149316	−	0.0862074i	−0.572796	+	0.819698i	$0.694141\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.13306	−	1.80887i	0.205253	−	0.118503i	−0.393850	−	0.919175i	$-0.628857\pi$
$233$	3.13306	−	1.80887i	0.205253	−	0.118503i	0.599103	+	0.800672i	$0.295524\pi$
$234$		0			0
$235$	−3.71743	+	6.43878i	−0.242499	+	0.420020i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.8069	+	7.39409i	−0.828412	+	0.478284i	−0.853309	−	0.521406i	$-0.825408\pi$
$239$	−12.8069	+	7.39409i	−0.828412	+	0.478284i	0.0248967	+	0.999690i	$0.492074\pi$
$240$		0			0
$241$	−5.57890	+	3.22098i	−0.359368	+	0.207481i	−0.668804	−	0.743439i	$-0.733193\pi$
$241$	−5.57890	+	3.22098i	−0.359368	+	0.207481i	0.309435	+	0.950921i	$0.399860\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−13.5267	−	12.1071i	−0.864187	−	0.773492i
$246$		0			0
$247$	−3.85909	+	6.68413i	−0.245548	+	0.425301i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.21012			0.391979			0.195990	−	0.980606i	$-0.437208\pi$
$251$	6.21012			0.391979			0.195990	+	0.980606i	$0.437208\pi$
$252$		0			0
$253$	11.9638			0.752158
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.5883	+	18.3395i	−0.660482	+	1.14399i	0.320007	+	0.947415i	$0.396315\pi$
$257$	−10.5883	+	18.3395i	−0.660482	+	1.14399i	−0.980489	+	0.196573i	$0.937019\pi$
$258$		0			0
$259$	6.56089	−	17.1677i	0.407674	−	1.06675i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	14.8592	−	8.57897i	0.916258	−	0.529002i	0.0338189	−	0.999428i	$-0.489233\pi$
$263$	14.8592	−	8.57897i	0.916258	−	0.529002i	0.882439	+	0.470426i	$0.155900\pi$
$264$		0			0
$265$	−20.2650	+	11.7000i	−1.24487	+	0.718725i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.8061	−	18.7166i	0.658857	−	1.14117i	−0.322055	−	0.946721i	$-0.604373\pi$
$269$	10.8061	−	18.7166i	0.658857	−	1.14117i	0.980912	−	0.194453i	$-0.0622932\pi$
$270$		0			0
$271$	21.8475	−	12.6137i	1.32714	−	0.766227i	0.342287	−	0.939595i	$-0.388798\pi$
$271$	21.8475	−	12.6137i	1.32714	−	0.766227i	0.984857	+	0.173368i	$0.0554652\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.29454	+	3.05680i	0.319273	+	0.184332i
$276$		0			0
$277$	−11.2256	−	19.4434i	−0.674482	−	1.16824i	−0.976620	−	0.214973i	$-0.931033\pi$
$277$	−11.2256	−	19.4434i	−0.674482	−	1.16824i	0.302137	−	0.953264i	$-0.402300\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−10.3691	−	5.98662i	−0.618571	−	0.357132i	0.157742	−	0.987480i	$-0.449579\pi$
$281$	−10.3691	−	5.98662i	−0.618571	−	0.357132i	−0.776312	+	0.630349i	$0.782912\pi$
$282$		0			0
$283$			21.7299i			1.29171i	0.763460	+	0.645855i	$0.223499\pi$
$283$			21.7299i			1.29171i	−0.763460	+	0.645855i	$0.776501\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	11.4883	−	30.0612i	0.678134	−	1.77445i
$288$		0			0
$289$	8.02442	+	13.8987i	0.472024	+	0.817570i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5.38110	−	9.32035i	−0.314367	−	0.544500i	0.664935	−	0.746901i	$-0.268459\pi$
$293$	−5.38110	−	9.32035i	−0.314367	−	0.544500i	−0.979303	+	0.202400i	$0.935126\pi$
$294$		0			0
$295$	−0.569833	+	0.986980i	−0.0331770	+	0.0574642i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.6873			0.618063
$300$		0			0
$301$	−28.6143	+	4.57554i	−1.64930	+	0.263730i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8.84966	+	5.10935i	0.506730	+	0.292561i
$306$		0			0
$307$		−	4.22117i		−	0.240915i	−0.992719	−	0.120457i	$-0.961564\pi$
$307$		−	4.22117i		−	0.240915i	0.992719	−	0.120457i	$-0.0384361\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.18531			0.237327			0.118664	−	0.992935i	$-0.462139\pi$
$311$	4.18531			0.237327			0.118664	+	0.992935i	$0.462139\pi$
$312$		0			0
$313$			22.0395i			1.24574i	0.782324	+	0.622872i	$0.214034\pi$
$313$			22.0395i			1.24574i	−0.782324	+	0.622872i	$0.785966\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	9.71612i		−	0.545711i	−0.962055	−	0.272856i	$-0.912032\pi$
$317$		−	9.71612i		−	0.545711i	0.962055	−	0.272856i	$-0.0879682\pi$
$318$		0			0
$319$	1.09496			0.0613061
$320$		0			0
$321$		0			0
$322$		0			0
$323$			2.37834i			0.132335i
$324$		0			0
$325$	4.72963	+	2.73065i	0.262353	+	0.151469i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.78215	+	5.88761i	0.263649	+	0.324594i
$330$		0			0
$331$	27.7176			1.52350			0.761749	−	0.647872i	$-0.224341\pi$
$331$	27.7176			1.52350			0.761749	+	0.647872i	$0.224341\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.73447	+	8.20034i	−0.258672	+	0.448032i
$336$		0			0
$337$	6.29141	+	10.8970i	0.342715	+	0.593600i	0.984936	−	0.172920i	$-0.0553201\pi$
$337$	6.29141	+	10.8970i	0.342715	+	0.593600i	−0.642221	+	0.766520i	$0.721987\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8.91788	+	15.4462i	0.482930	+	0.836460i
$342$		0			0
$343$	−16.4641	+	8.48130i	−0.888979	+	0.457947i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	16.7223i		−	0.897700i	−0.893607	−	0.448850i	$-0.851834\pi$
$347$		−	16.7223i		−	0.897700i	0.893607	−	0.448850i	$-0.148166\pi$
$348$		0			0
$349$	13.9900	+	8.07714i	0.748869	+	0.432359i	0.825285	−	0.564717i	$-0.191015\pi$
$349$	13.9900	+	8.07714i	0.748869	+	0.432359i	−0.0764164	+	0.997076i	$0.524348\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.82831	−	13.5590i	−0.416659	−	0.721674i	0.578942	−	0.815369i	$-0.303466\pi$
$353$	−7.82831	−	13.5590i	−0.416659	−	0.721674i	−0.995601	+	0.0936943i	$0.970132\pi$
$354$		0			0
$355$	−11.7923	−	6.80830i	−0.625872	−	0.361347i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	19.1396	−	11.0502i	1.01015	−	0.583209i	0.0989127	−	0.995096i	$-0.468464\pi$
$359$	19.1396	−	11.0502i	1.01015	−	0.583209i	0.911235	+	0.411887i	$0.135130\pi$
$360$		0			0
$361$	−6.52655	+	11.3043i	−0.343502	+	0.594964i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	36.5075	−	21.0776i	1.91089	−	1.10325i
$366$		0			0
$367$	−11.3893	+	6.57564i	−0.594519	+	0.343246i	−0.766882	−	0.641788i	$-0.778193\pi$
$367$	−11.3893	+	6.57564i	−0.594519	+	0.343246i	0.172363	+	0.985033i	$0.444860\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.76944	+	23.5732i	0.195700	+	1.22386i
$372$		0			0
$373$	2.21915	−	3.84367i	0.114903	−	0.199018i	−0.802838	−	0.596197i	$-0.796678\pi$
$373$	2.21915	−	3.84367i	0.114903	−	0.199018i	0.917741	+	0.397179i	$0.130011\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.978132			0.0503763
$378$		0			0
$379$	14.8875			0.764718			0.382359	−	0.924014i	$-0.375112\pi$
$379$	14.8875			0.764718			0.382359	+	0.924014i	$0.375112\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.81901	+	4.88267i	−0.144045	+	0.249493i	−0.929016	−	0.370039i	$-0.879344\pi$
$383$	−2.81901	+	4.88267i	−0.144045	+	0.249493i	0.784971	+	0.619532i	$0.212678\pi$
$384$		0			0
$385$	18.8696	−	15.3267i	0.961686	−	0.781121i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−33.3523	+	19.2560i	−1.69103	+	0.976317i	−0.737342	+	0.675519i	$0.763919\pi$
$389$	−33.3523	+	19.2560i	−1.69103	+	0.976317i	−0.953688	+	0.300797i	$0.902747\pi$
$390$		0			0
$391$	2.85206	−	1.64664i	0.144235	−	0.0832741i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−9.06795	+	15.7061i	−0.456258	+	0.790262i
$396$		0			0
$397$	2.50780	−	1.44788i	0.125863	−	0.0726669i	−0.435747	−	0.900069i	$-0.643516\pi$
$397$	2.50780	−	1.44788i	0.125863	−	0.0726669i	0.561609	+	0.827402i	$0.310182\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	22.2857	+	12.8666i	1.11289	+	0.642530i	0.939577	−	0.342337i	$-0.111218\pi$
$401$	22.2857	+	12.8666i	1.11289	+	0.642530i	0.173316	+	0.984866i	$0.444552\pi$
$402$		0			0
$403$	7.96637	+	13.7981i	0.396833	+	0.687335i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	21.3140	+	12.3057i	1.05650	+	0.609969i
$408$		0			0
$409$		−	28.8575i		−	1.42691i	−0.700699	−	0.713457i	$-0.747129\pi$
$409$		−	28.8575i		−	1.42691i	0.700699	−	0.713457i	$-0.252871\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.733041	+	0.902492i	0.0360706	+	0.0444088i
$414$		0			0
$415$	−18.7721	−	32.5142i	−0.921486	−	1.59606i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.07870	+	10.5286i	0.296964	+	0.514357i	0.975440	−	0.220266i	$-0.0706925\pi$
$419$	6.07870	+	10.5286i	0.296964	+	0.514357i	−0.678476	+	0.734623i	$0.737359\pi$
$420$		0			0
$421$	−13.9914	+	24.2338i	−0.681898	+	1.18108i	0.292503	+	0.956265i	$0.405512\pi$
$421$	−13.9914	+	24.2338i	−0.681898	+	1.18108i	−0.974401	+	0.224817i	$0.927821\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.68289			0.0816323
$426$		0			0
$427$	8.09211	−	6.57274i	0.391605	−	0.318077i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	13.9975	+	8.08147i	0.674236	+	0.389271i	0.797680	−	0.603081i	$-0.206060\pi$
$431$	13.9975	+	8.08147i	0.674236	+	0.389271i	−0.123444	+	0.992352i	$0.539394\pi$
$432$		0			0
$433$			5.27197i			0.253355i	0.991944	+	0.126677i	$0.0404313\pi$
$433$			5.27197i			0.253355i	−0.991944	+	0.126677i	$0.959569\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.23464			−0.393916
$438$		0			0
$439$		−	10.5108i		−	0.501654i	−0.968032	−	0.250827i	$-0.919298\pi$
$439$		−	10.5108i		−	0.501654i	0.968032	−	0.250827i	$-0.0807025\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	5.95601i		−	0.282978i	−0.989940	−	0.141489i	$-0.954811\pi$
$443$		−	5.95601i		−	0.282978i	0.989940	−	0.141489i	$-0.0451891\pi$
$444$		0			0
$445$	−11.9866			−0.568219
$446$		0			0
$447$		0			0
$448$		0			0
$449$			1.85897i			0.0877304i	0.999037	+	0.0438652i	$0.0139672\pi$
$449$			1.85897i			0.0877304i	−0.999037	+	0.0438652i	$0.986033\pi$
$450$		0			0
$451$	37.3215	+	21.5476i	1.75740	+	1.01464i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	16.8563	−	13.6914i	0.790236	−	0.641862i
$456$		0			0
$457$	29.7493			1.39162			0.695808	−	0.718228i	$-0.255047\pi$
$457$	29.7493			1.39162			0.695808	+	0.718228i	$0.255047\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−14.9738	+	25.9353i	−0.697398	+	1.20793i	0.271968	+	0.962306i	$0.412325\pi$
$461$	−14.9738	+	25.9353i	−0.697398	+	1.20793i	−0.969366	+	0.245622i	$0.921008\pi$
$462$		0			0
$463$	7.57406	+	13.1187i	0.351997	+	0.609676i	0.986599	−	0.163163i	$-0.0521695\pi$
$463$	7.57406	+	13.1187i	0.351997	+	0.609676i	−0.634603	+	0.772839i	$0.718836\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.85158	+	10.1352i	0.270779	+	0.469002i	0.969061	−	0.246820i	$-0.0793855\pi$
$467$	5.85158	+	10.1352i	0.270779	+	0.469002i	−0.698283	+	0.715822i	$0.746052\pi$
$468$		0			0
$469$	6.09048	+	7.49837i	0.281232	+	0.346243i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	38.8050i		−	1.78425i
$474$		0			0
$475$	−3.64421	−	2.10399i	−0.167208	−	0.0965375i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.07088	−	10.5151i	−0.277385	−	0.480446i	0.693349	−	0.720602i	$-0.256135\pi$
$479$	−6.07088	−	10.5151i	−0.277385	−	0.480446i	−0.970734	+	0.240156i	$0.922801\pi$
$480$		0			0
$481$	19.0399	+	10.9927i	0.868144	+	0.501223i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	32.3664	−	18.6868i	1.46968	−	0.848522i
$486$		0			0
$487$	−5.22070	+	9.04251i	−0.236572	+	0.409755i	−0.959728	−	0.280929i	$-0.909357\pi$
$487$	−5.22070	+	9.04251i	−0.236572	+	0.409755i	0.723156	+	0.690685i	$0.242691\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−20.9972	+	12.1227i	−0.947589	+	0.547091i	−0.892331	−	0.451382i	$-0.850931\pi$
$491$	−20.9972	+	12.1227i	−0.947589	+	0.547091i	−0.0552576	+	0.998472i	$0.517598\pi$
$492$		0			0
$493$	0.261029	−	0.150705i	0.0117561	−	0.00678741i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−10.7829	+	8.75829i	−0.483678	+	0.392863i
$498$		0			0
$499$	0.174119	−	0.301584i	0.00779466	−	0.0135007i	−0.862102	−	0.506735i	$-0.830852\pi$
$499$	0.174119	−	0.301584i	0.00779466	−	0.0135007i	0.869896	+	0.493234i	$0.164186\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−32.3099			−1.44063			−0.720314	−	0.693648i	$-0.756002\pi$
$503$	−32.3099			−1.44063			−0.720314	+	0.693648i	$0.756002\pi$
$504$		0			0
$505$	−22.3787			−0.995839
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.85232	−	11.8686i	0.303724	−	0.526065i	−0.673253	−	0.739413i	$-0.735103\pi$
$509$	6.85232	−	11.8686i	0.303724	−	0.526065i	0.976976	+	0.213347i	$0.0684366\pi$
$510$		0			0
$511$	−6.79069	−	42.4673i	−0.300402	−	1.87864i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	13.6941	−	7.90630i	0.603434	−	0.348393i
$516$		0			0
$517$	−8.79650	+	5.07866i	−0.386870	+	0.223359i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8.73516	+	15.1297i	−0.382694	+	0.662846i	−0.991446	−	0.130514i	$-0.958337\pi$
$521$	−8.73516	+	15.1297i	−0.382694	+	0.662846i	0.608752	+	0.793361i	$0.291670\pi$
$522$		0			0
$523$	−23.3238	+	13.4660i	−1.01988	+	0.588827i	−0.914069	−	0.405559i	$-0.867077\pi$
$523$	−23.3238	+	13.4660i	−1.01988	+	0.588827i	−0.105810	+	0.994386i	$0.533743\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.25188	+	2.45482i	0.185215	+	0.106934i
$528$		0			0
$529$	−5.79877	−	10.0438i	−0.252120	−	0.436685i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	33.3394	+	19.2485i	1.44409	+	0.833746i
$534$		0			0
$535$		−	41.5768i		−	1.79752i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7.73379	−	23.5643i	−0.333118	−	1.01499i
$540$		0			0
$541$	−7.43375	−	12.8756i	−0.319602	−	0.553567i	0.660803	−	0.750559i	$-0.270216\pi$
$541$	−7.43375	−	12.8756i	−0.319602	−	0.553567i	−0.980405	+	0.196993i	$0.936882\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.22666	+	12.5169i	0.309556	+	0.536167i
$546$		0			0
$547$	−4.17953	+	7.23915i	−0.178704	+	0.309524i	−0.941437	−	0.337190i	$-0.890524\pi$
$547$	−4.17953	+	7.23915i	−0.178704	+	0.309524i	0.762733	+	0.646713i	$0.223857\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.753658			−0.0321069
$552$		0			0
$553$	11.6651	+	14.3617i	0.496052	+	0.610720i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.617917	−	0.356754i	−0.0261820	−	0.0151162i	0.486852	−	0.873485i	$-0.338145\pi$
$557$	−0.617917	−	0.356754i	−0.0261820	−	0.0151162i	−0.513034	+	0.858368i	$0.671478\pi$
$558$		0			0
$559$		−	34.6646i		−	1.46616i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−27.1877			−1.14582			−0.572912	−	0.819617i	$-0.694187\pi$
$563$	−27.1877			−1.14582			−0.572912	+	0.819617i	$0.694187\pi$
$564$		0			0
$565$			42.5659i			1.79076i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	1.75321i		−	0.0734985i	−0.999325	−	0.0367492i	$-0.988300\pi$
$569$		−	1.75321i		−	0.0734985i	0.999325	−	0.0367492i	$-0.0117003\pi$
$570$		0			0
$571$	0.457634			0.0191514			0.00957570	−	0.999954i	$-0.496952\pi$
$571$	0.457634			0.0191514			0.00957570	+	0.999954i	$0.496952\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			5.82675i			0.242992i
$576$		0			0
$577$	19.8800	+	11.4777i	0.827617	+	0.477825i	0.853036	−	0.521852i	$-0.174759\pi$
$577$	19.8800	+	11.4777i	0.827617	+	0.477825i	−0.0254193	+	0.999677i	$0.508092\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−37.8221	+	6.04790i	−1.56912	+	0.250909i
$582$		0			0
$583$	−31.9685			−1.32400
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.81513	−	4.87594i	0.116193	−	0.201252i	−0.802063	−	0.597239i	$-0.796264\pi$
$587$	2.81513	−	4.87594i	0.116193	−	0.201252i	0.918256	+	0.395987i	$0.129598\pi$
$588$		0			0
$589$	−6.13814	−	10.6316i	−0.252918	−	0.438066i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.35121	+	2.34037i	0.0554876	+	0.0961074i	0.892435	−	0.451176i	$-0.148995\pi$
$593$	1.35121	+	2.34037i	0.0554876	+	0.0961074i	−0.836947	+	0.547283i	$0.815662\pi$
$594$		0			0
$595$	2.38886	−	6.25086i	0.0979337	−	0.256260i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			11.9966i			0.490168i	0.969502	+	0.245084i	$0.0788156\pi$
$599$			11.9966i			0.490168i	−0.969502	+	0.245084i	$0.921184\pi$
$600$		0			0
$601$	−26.7530	−	15.4459i	−1.09128	−	0.630050i	−0.157363	−	0.987541i	$-0.550299\pi$
$601$	−26.7530	−	15.4459i	−1.09128	−	0.630050i	−0.933917	+	0.357491i	$0.883632\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.01348	+	3.48746i	0.0818598	+	0.141785i
$606$		0			0
$607$	5.95361	+	3.43732i	0.241649	+	0.139516i	0.615935	−	0.787797i	$-0.288779\pi$
$607$	5.95361	+	3.43732i	0.241649	+	0.139516i	−0.374285	+	0.927314i	$0.622112\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.85794	+	4.53678i	−0.317898	+	0.183539i
$612$		0			0
$613$	−16.6455	+	28.8308i	−0.672304	+	1.16447i	0.304945	+	0.952370i	$0.401362\pi$
$613$	−16.6455	+	28.8308i	−0.672304	+	1.16447i	−0.977249	+	0.212095i	$0.931971\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−18.0757	+	10.4360i	−0.727702	+	0.420139i	−0.817581	−	0.575814i	$-0.804685\pi$
$617$	−18.0757	+	10.4360i	−0.727702	+	0.420139i	0.0898790	+	0.995953i	$0.471352\pi$
$618$		0			0
$619$	−16.7876	+	9.69230i	−0.674749	+	0.389567i	−0.797874	−	0.602825i	$-0.794042\pi$
$619$	−16.7876	+	9.69230i	−0.674749	+	0.389567i	0.123125	+	0.992391i	$0.460709\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.36546	+	11.4230i	−0.174898	+	0.457652i
$624$		0			0
$625$	15.3251	−	26.5439i	0.613005	−	1.06175i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.77476			0.270127
$630$		0			0
$631$	−27.3629			−1.08930			−0.544649	−	0.838664i	$-0.683337\pi$
$631$	−27.3629			−1.08930			−0.544649	+	0.838664i	$0.683337\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	23.3734	−	40.4839i	0.927546	−	1.60656i
$636$		0			0
$637$	−6.90861	−	21.0500i	−0.273729	−	0.834033i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	30.0686	−	17.3601i	1.18764	−	0.685683i	0.229868	−	0.973222i	$-0.426170\pi$
$641$	30.0686	−	17.3601i	1.18764	−	0.685683i	0.957769	+	0.287539i	$0.0928371\pi$
$642$		0			0
$643$	−3.52980	+	2.03793i	−0.139202	+	0.0803681i	−0.567983	−	0.823040i	$-0.692276\pi$
$643$	−3.52980	+	2.03793i	−0.139202	+	0.0803681i	0.428782	+	0.903408i	$0.358943\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.53976	+	11.3272i	−0.257105	+	0.445318i	−0.965465	−	0.260533i	$-0.916102\pi$
$647$	−6.53976	+	11.3272i	−0.257105	+	0.445318i	0.708360	+	0.705851i	$0.249435\pi$
$648$		0			0
$649$	−1.34839	+	0.778492i	−0.0529288	+	0.0305585i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.40555	+	3.12090i	0.211535	+	0.122130i	0.602025	−	0.798477i	$-0.294361\pi$
$653$	5.40555	+	3.12090i	0.211535	+	0.122130i	−0.390489	+	0.920607i	$0.627694\pi$
$654$		0			0
$655$	−13.5365	−	23.4459i	−0.528916	−	0.916109i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.79139	−	1.03426i	−0.0697827	−	0.0402891i	0.464703	−	0.885467i	$-0.346161\pi$
$659$	−1.79139	−	1.03426i	−0.0697827	−	0.0402891i	−0.534485	+	0.845178i	$0.679495\pi$
$660$		0			0
$661$			15.6238i			0.607695i	0.952721	+	0.303848i	$0.0982714\pi$
$661$			15.6238i			0.607695i	−0.952721	+	0.303848i	$0.901729\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−12.9879	+	10.5493i	−0.503649	+	0.409085i
$666$		0			0
$667$	0.521792	+	0.903771i	0.0202039	+	0.0349941i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6.98027	+	12.0902i	0.269470	+	0.466736i
$672$		0			0
$673$	−15.1108	+	26.1727i	−0.582478	+	1.00888i	0.412706	+	0.910864i	$0.364584\pi$
$673$	−15.1108	+	26.1727i	−0.582478	+	1.00888i	−0.995185	+	0.0980180i	$0.968750\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−28.6876			−1.10256			−0.551278	−	0.834322i	$-0.685859\pi$
$677$	−28.6876			−1.10256			−0.551278	+	0.834322i	$0.685859\pi$
$678$		0			0
$679$	−6.02040	−	37.6501i	−0.231042	−	1.44488i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	7.48345	+	4.32057i	0.286346	+	0.165322i	0.636293	−	0.771448i	$-0.280467\pi$
$683$	7.48345	+	4.32057i	0.286346	+	0.165322i	−0.349947	+	0.936770i	$0.613800\pi$
$684$		0			0
$685$			9.98423i			0.381478i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−28.5575			−1.08796
$690$		0			0
$691$		−	12.2390i		−	0.465594i	−0.972525	−	0.232797i	$-0.925212\pi$
$691$		−	12.2390i		−	0.465594i	0.972525	−	0.232797i	$-0.0747878\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	56.1881i		−	2.13134i
$696$		0			0
$697$	11.8628			0.449336
$698$		0			0
$699$		0			0
$700$		0			0
$701$			3.05683i			0.115455i	0.998332	+	0.0577274i	$0.0183854\pi$
$701$			3.05683i			0.115455i	−0.998332	+	0.0577274i	$0.981615\pi$
$702$		0			0
$703$	−14.6704	−	8.46994i	−0.553303	−	0.319450i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.15021	+	21.3264i	−0.306520	+	0.802063i
$708$		0			0
$709$	−12.0820			−0.453750			−0.226875	−	0.973924i	$-0.572851\pi$
$709$	−12.0820			−0.453750			−0.226875	+	0.973924i	$0.572851\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−8.49945	+	14.7215i	−0.318307	+	0.551324i
$714$		0			0
$715$	14.5403	+	25.1845i	0.543776	+	0.941847i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.27436	−	10.8675i	−0.233994	−	0.405290i	0.724986	−	0.688764i	$-0.241846\pi$
$719$	−6.27436	−	10.8675i	−0.233994	−	0.405290i	−0.958980	+	0.283474i	$0.908513\pi$
$720$		0			0
$721$	−2.54721	−	15.9296i	−0.0948630	−	0.593250i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			0.533281i			0.0198056i
$726$		0			0
$727$	−16.2736	−	9.39555i	−0.603553	−	0.348462i	0.166885	−	0.985976i	$-0.446629\pi$
$727$	−16.2736	−	9.39555i	−0.603553	−	0.348462i	−0.770438	+	0.637515i	$0.779962\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.34092	−	9.25075i	−0.197541	−	0.342151i
$732$		0			0
$733$	30.6628	+	17.7032i	1.13256	+	0.653882i	0.944577	−	0.328290i	$-0.106472\pi$
$733$	30.6628	+	17.7032i	1.13256	+	0.653882i	0.187981	+	0.982173i	$0.439806\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.2031	+	6.46811i	−0.412671	+	0.238256i
$738$		0			0
$739$	11.1371	−	19.2900i	0.409684	−	0.709593i	−0.585170	−	0.810910i	$-0.698972\pi$
$739$	11.1371	−	19.2900i	0.409684	−	0.709593i	0.994854	+	0.101317i	$0.0323057\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8.68932	−	5.01678i	0.318780	−	0.184048i	−0.332069	−	0.943255i	$-0.607747\pi$
$743$	8.68932	−	5.01678i	0.318780	−	0.184048i	0.650849	+	0.759207i	$0.274413\pi$
$744$		0			0
$745$	21.3428	−	12.3222i	0.781938	−	0.451452i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−39.6218	−	15.1421i	−1.44775	−	0.553279i
$750$		0			0
$751$	−16.7771	+	29.0587i	−0.612203	+	1.06037i	0.378665	+	0.925534i	$0.376383\pi$
$751$	−16.7771	+	29.0587i	−0.612203	+	1.06037i	−0.990868	+	0.134833i	$0.956950\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−55.3373			−2.01393
$756$		0			0
$757$	0.0340541			0.00123772			0.000618859	−	1.00000i	$-0.499803\pi$
$757$	0.0340541			0.00123772			0.000618859		1.00000i	$0.499803\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.4846	+	19.8919i	−0.416315	+	0.721079i	−0.995566	−	0.0940705i	$-0.970012\pi$
$761$	−11.4846	+	19.8919i	−0.416315	+	0.721079i	0.579250	+	0.815150i	$0.303345\pi$
$762$		0			0
$763$	14.5603	−	2.32825i	0.527118	−	0.0842882i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.20452	+	0.695429i	−0.0434926	+	0.0251105i
$768$		0			0
$769$	−14.3775	+	8.30086i	−0.518466	+	0.299337i	−0.736307	−	0.676648i	$-0.763432\pi$
$769$	−14.3775	+	8.30086i	−0.518466	+	0.299337i	0.217841	+	0.975984i	$0.430099\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.803138	+	1.39107i	−0.0288868	+	0.0500335i	−0.880107	−	0.474775i	$-0.842530\pi$
$773$	−0.803138	+	1.39107i	−0.0288868	+	0.0500335i	0.851221	+	0.524808i	$0.175863\pi$
$774$		0			0
$775$	−7.52280	+	4.34329i	−0.270227	+	0.156016i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−25.6883	−	14.8311i	−0.920378	−	0.531380i
$780$		0			0
$781$	−9.30133	−	16.1104i	−0.332828	−	0.576475i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5.45988	−	3.15226i	−0.194871	−	0.112509i
$786$		0			0
$787$		−	15.9080i		−	0.567060i	−0.958963	−	0.283530i	$-0.908494\pi$
$787$		−	15.9080i		−	0.567060i	0.958963	−	0.283530i	$-0.0915055\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	40.5644	+	15.5023i	1.44231	+	0.551198i
$792$		0			0
$793$	6.23549	+	10.8002i	0.221429	+	0.383526i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−10.1844	−	17.6399i	−0.360749	−	0.624836i	0.627335	−	0.778749i	$-0.284146\pi$
$797$	−10.1844	−	17.6399i	−0.360749	−	0.624836i	−0.988084	+	0.153914i	$0.950812\pi$
$798$		0			0
$799$	−1.39800	+	2.42141i	−0.0494578	+	0.0856634i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	57.5915			2.03236
$804$		0			0
$805$	21.6426	+	8.27106i	0.762803	+	0.291516i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−35.7396	−	20.6343i	−1.25654	−	0.725462i	−0.284137	−	0.958784i	$-0.591707\pi$
$809$	−35.7396	−	20.6343i	−1.25654	−	0.725462i	−0.972400	+	0.233322i	$0.925040\pi$
$810$		0			0
$811$		−	19.3660i		−	0.680032i	−0.940420	−	0.340016i	$-0.889568\pi$
$811$		−	19.3660i		−	0.680032i	0.940420	−	0.340016i	$-0.110432\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−43.1055			−1.50992
$816$		0			0
$817$			26.7093i			0.934441i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			32.4939i			1.13404i	0.823703	+	0.567022i	$0.191905\pi$
$821$			32.4939i			1.13404i	−0.823703	+	0.567022i	$0.808095\pi$
$822$		0			0
$823$	55.7913			1.94476			0.972381	−	0.233397i	$-0.0749843\pi$
$823$	55.7913			1.94476			0.972381	+	0.233397i	$0.0749843\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			18.2989i			0.636314i	0.948038	+	0.318157i	$0.103064\pi$
$827$			18.2989i			0.636314i	−0.948038	+	0.318157i	$0.896936\pi$
$828$		0			0
$829$	12.6483	+	7.30247i	0.439292	+	0.253625i	0.703297	−	0.710896i	$-0.251710\pi$
$829$	12.6483	+	7.30247i	0.439292	+	0.253625i	−0.264005	+	0.964521i	$0.585044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5.08693	−	4.55307i	−0.176252	−	0.157754i
$834$		0			0
$835$	−54.9983			−1.90330
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15.7379	−	27.2588i	0.543332	−	0.941079i	−0.455378	−	0.890298i	$-0.650496\pi$
$839$	15.7379	−	27.2588i	0.543332	−	0.941079i	0.998710	−	0.0507803i	$-0.0161708\pi$
$840$		0			0
$841$	−14.4522	−	25.0320i	−0.498353	−	0.863173i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.86801	−	6.69959i	−0.133064	−	0.230473i
$846$		0			0
$847$	4.05677	−	0.648694i	0.139392	−	0.0222894i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			23.4565i			0.804080i
$852$		0			0
$853$	21.2554	+	12.2718i	0.727771	+	0.420179i	0.817606	−	0.575778i	$-0.195301\pi$
$853$	21.2554	+	12.2718i	0.727771	+	0.420179i	−0.0898353	+	0.995957i	$0.528634\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5.86511	+	10.1587i	0.200348	+	0.347014i	0.948641	−	0.316355i	$-0.102459\pi$
$857$	5.86511	+	10.1587i	0.200348	+	0.347014i	−0.748292	+	0.663369i	$0.769126\pi$
$858$		0			0
$859$	−29.5421	−	17.0561i	−1.00796	−	0.581948i	−0.0973688	−	0.995248i	$-0.531043\pi$
$859$	−29.5421	−	17.0561i	−1.00796	−	0.581948i	−0.910595	+	0.413300i	$0.864376\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−20.8863	+	12.0587i	−0.710978	+	0.410483i	−0.811423	−	0.584459i	$-0.801307\pi$
$863$	−20.8863	+	12.0587i	−0.710978	+	0.410483i	0.100445	+	0.994943i	$0.467973\pi$
$864$		0			0
$865$	−13.2018	+	22.8663i	−0.448876	+	0.777477i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−21.4573	+	12.3884i	−0.727890	+	0.420248i
$870$		0			0
$871$	−10.0078	+	5.77798i	−0.339100	+	0.195779i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−14.1650	−	17.4394i	−0.478865	−	0.589560i
$876$		0			0
$877$	25.8753	−	44.8173i	0.873747	−	1.51337i	0.0156551	−	0.999877i	$-0.495017\pi$
$877$	25.8753	−	44.8173i	0.873747	−	1.51337i	0.858092	−	0.513496i	$-0.171650\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30.3925			−1.02395			−0.511975	−	0.859000i	$-0.671086\pi$
$881$	−30.3925			−1.02395			−0.511975	+	0.859000i	$0.671086\pi$
$882$		0			0
$883$	35.4897			1.19432			0.597162	−	0.802121i	$-0.296295\pi$
$883$	35.4897			1.19432			0.597162	+	0.802121i	$0.296295\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	23.2756	−	40.3145i	0.781517	−	1.35363i	−0.149540	−	0.988756i	$-0.547779\pi$
$887$	23.2756	−	40.3145i	0.781517	−	1.35363i	0.931058	−	0.364872i	$-0.118887\pi$
$888$		0			0
$889$	−30.0679	−	37.0184i	−1.00844	−	1.24156i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6.05460	−	3.49562i	0.202609	−	0.116977i
$894$		0			0
$895$	1.52874	−	0.882620i	0.0511003	−	0.0295027i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.777893	+	1.34735i	−0.0259442	+	0.0449366i
$900$		0			0
$901$	−7.62099	+	4.39998i	−0.253892	+	0.146585i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	42.9641	+	24.8053i	1.42817	+	0.824556i
$906$		0			0
$907$	7.31653	+	12.6726i	0.242941	+	0.420787i	0.961551	−	0.274627i	$-0.0885544\pi$
$907$	7.31653	+	12.6726i	0.242941	+	0.420787i	−0.718610	+	0.695414i	$0.755221\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6.67661	+	3.85474i	0.221206	+	0.127713i	0.606508	−	0.795077i	$-0.292570\pi$
$911$	6.67661	+	3.85474i	0.221206	+	0.127713i	−0.385303	+	0.922790i	$0.625903\pi$
$912$		0			0
$913$		−	51.2919i		−	1.69752i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−27.2734	+	4.36113i	−0.900648	+	0.144017i
$918$		0			0
$919$	13.4738	+	23.3373i	0.444460	+	0.769827i	0.998014	−	0.0629860i	$-0.0200623\pi$
$919$	13.4738	+	23.3373i	0.444460	+	0.769827i	−0.553555	+	0.832813i	$0.686729\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−8.30890	−	14.3914i	−0.273491	−	0.473700i
$924$		0			0
$925$	−5.99325	+	10.3806i	−0.197057	+	0.341312i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−27.6274			−0.906427			−0.453213	−	0.891402i	$-0.649722\pi$
$929$	−27.6274			−0.906427			−0.453213	+	0.891402i	$0.649722\pi$
$930$		0			0
$931$	5.32314	+	16.2192i	0.174459	+	0.531563i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	7.76057	+	4.48057i	0.253798	+	0.146530i
$936$		0			0
$937$			7.60743i			0.248524i	0.992249	+	0.124262i	$0.0396563\pi$
$937$			7.60743i			0.248524i	−0.992249	+	0.124262i	$0.960344\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	41.2996			1.34633			0.673165	−	0.739493i	$-0.264934\pi$
$941$	41.2996			1.34633			0.673165	+	0.739493i	$0.264934\pi$
$942$		0			0
$943$			41.0731i			1.33753i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			41.6059i			1.35201i	0.736897	+	0.676005i	$0.236290\pi$
$947$			41.6059i			1.35201i	−0.736897	+	0.676005i	$0.763710\pi$
$948$		0			0
$949$	51.4466			1.67003
$950$		0			0
$951$		0			0
$952$		0			0
$953$			47.2656i			1.53108i	0.643386	+	0.765542i	$0.277529\pi$
$953$			47.2656i			1.53108i	−0.643386	+	0.765542i	$0.722471\pi$
$954$		0			0
$955$	47.5038	+	27.4263i	1.53719	+	0.887494i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	9.51477	+	3.63621i	0.307248	+	0.117419i
$960$		0			0
$961$	5.65789			0.182512
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16.6344	+	28.8116i	−0.535480	+	0.927479i
$966$		0			0
$967$	−2.60192	−	4.50665i	−0.0836720	−	0.144924i	0.821153	−	0.570709i	$-0.193331\pi$
$967$	−2.60192	−	4.50665i	−0.0836720	−	0.144924i	−0.904825	+	0.425785i	$0.859998\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	8.67233	+	15.0209i	0.278308	+	0.482044i	0.970964	−	0.239224i	$-0.0768929\pi$
$971$	8.67233	+	15.0209i	0.278308	+	0.482044i	−0.692656	+	0.721268i	$0.743560\pi$
$972$		0			0
$973$	−53.5461	−	20.4634i	−1.71661	−	0.656027i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			26.9670i			0.862751i	0.902172	+	0.431376i	$0.141972\pi$
$977$			26.9670i			0.862751i	−0.902172	+	0.431376i	$0.858028\pi$
$978$		0			0
$979$	−14.1818	−	8.18789i	−0.453254	−	0.261686i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	8.55646	+	14.8202i	0.272909	+	0.472692i	0.969605	−	0.244674i	$-0.0786810\pi$
$983$	8.55646	+	14.8202i	0.272909	+	0.472692i	−0.696697	+	0.717366i	$0.745348\pi$
$984$		0			0
$985$	−50.0512	−	28.8971i	−1.59476	−	0.920738i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	32.0293	−	18.4921i	1.01847	−	0.588015i
$990$		0			0
$991$	−5.84386	+	10.1219i	−0.185636	+	0.321531i	−0.943791	−	0.330544i	$-0.892768\pi$
$991$	−5.84386	+	10.1219i	−0.185636	+	0.321531i	0.758155	+	0.652075i	$0.226101\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−17.2624	+	9.96646i	−0.547255	+	0.315958i
$996$		0			0
$997$	−25.0296	+	14.4508i	−0.792695	+	0.457663i	−0.840910	−	0.541174i	$-0.817980\pi$
$997$	−25.0296	+	14.4508i	−0.792695	+	0.457663i	0.0482154	+	0.998837i	$0.484647\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.bs.a.521.5		48
3.2	odd	2		504.2.bs.a.353.16	yes	48
4.3	odd	2		3024.2.ca.e.2033.5		48
7.5	odd	6		1512.2.cx.a.89.5		48
9.4	even	3		504.2.cx.a.185.24	yes	48
9.5	odd	6		1512.2.cx.a.17.5		48
12.11	even	2		1008.2.ca.e.353.9		48
21.5	even	6		504.2.cx.a.425.24	yes	48
28.19	even	6		3024.2.df.e.1601.5		48
36.23	even	6		3024.2.df.e.17.5		48
36.31	odd	6		1008.2.df.e.689.1		48
63.5	even	6	inner	1512.2.bs.a.1097.5		48
63.40	odd	6		504.2.bs.a.257.16	✓	48
84.47	odd	6		1008.2.df.e.929.1		48
252.103	even	6		1008.2.ca.e.257.9		48
252.131	odd	6		3024.2.ca.e.2609.5		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.bs.a.257.16	✓	48	63.40	odd	6
504.2.bs.a.353.16	yes	48	3.2	odd	2
504.2.cx.a.185.24	yes	48	9.4	even	3
504.2.cx.a.425.24	yes	48	21.5	even	6
1008.2.ca.e.257.9		48	252.103	even	6
1008.2.ca.e.353.9		48	12.11	even	2
1008.2.df.e.689.1		48	36.31	odd	6
1008.2.df.e.929.1		48	84.47	odd	6
1512.2.bs.a.521.5		48	1.1	even	1	trivial
1512.2.bs.a.1097.5		48	63.5	even	6	inner
1512.2.cx.a.17.5		48	9.5	odd	6
1512.2.cx.a.89.5		48	7.5	odd	6
3024.2.ca.e.2033.5		48	4.3	odd	2
3024.2.ca.e.2609.5		48	252.131	odd	6
3024.2.df.e.17.5		48	36.23	even	6
3024.2.df.e.1601.5		48	28.19	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:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.bs (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.29668 + 2.24592i) q^{5} +(1.66807 + 2.05367i) q^{7} +(-3.06832 + 1.77150i) q^{11} +(-2.74094 + 1.58248i) q^{13} +(-0.487640 + 0.844616i) q^{17} +(2.11191 - 1.21931i) q^{19} +(-2.92435 - 1.68838i) q^{23} +(-0.862775 - 1.49437i) q^{25} +(-0.267645 - 0.154525i) q^{29} -5.03410i q^{31} +(-6.77533 + 1.08340i) q^{35} +(-3.47324 - 6.01583i) q^{37} +(-6.08175 - 10.5339i) q^{41} +(-5.47630 + 9.48523i) q^{43} +2.86688 q^{47} +(-1.43508 + 6.85132i) q^{49} +(7.81416 + 4.51151i) q^{53} -9.18828i q^{55} +0.439455 q^{59} -3.94033i q^{61} -8.20791i q^{65} +3.65121 q^{67} +5.25055i q^{71} +(-14.0773 - 8.12752i) q^{73} +(-8.75624 - 3.34632i) q^{77} +6.99318 q^{79} +(-7.23851 + 12.5375i) q^{83} +(-1.26463 - 2.19040i) q^{85} +(2.31101 + 4.00279i) q^{89} +(-7.82197 - 2.98928i) q^{91} +6.32426i q^{95} +(-12.4805 - 7.20560i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 12 q^{23} - 24 q^{25} - 18 q^{29} - 6 q^{41} - 6 q^{43} - 36 q^{47} + 6 q^{49} - 12 q^{53} + 36 q^{77} - 12 q^{79} - 18 q^{89} + 6 q^{91}+O(q^{100}) \) 48 * q + 12 * q^23 - 24 * q^25 - 18 * q^29 - 6 * q^41 - 6 * q^43 - 36 * q^47 + 6 * q^49 - 12 * q^53 + 36 * q^77 - 12 * q^79 - 18 * q^89 + 6 * q^91

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