LMFDB - Embedded newform 2268.2.bm.j.1025.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(593,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.593"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.bm (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,4,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1025.6
Character	$\chi$	$=$	2268.1025
Dual form			2268.2.bm.j.593.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.13332 q^{5} +(-1.77285 - 1.96392i) q^{7} +2.65687i q^{11} +(1.59436 + 0.920505i) q^{13} +(0.267475 - 0.463281i) q^{17} +(2.57178 - 1.48482i) q^{19} +0.969928i q^{23} -3.71559 q^{25} +(-2.66955 + 1.54127i) q^{29} +(-0.682710 + 0.394163i) q^{31} +(2.00920 + 2.22575i) q^{35} +(-1.73316 - 3.00191i) q^{37} +(2.22822 - 3.85939i) q^{41} +(-0.849465 - 1.47132i) q^{43} +(-5.14286 + 8.90769i) q^{47} +(-0.713980 + 6.96349i) q^{49} +(11.9002 + 6.87059i) q^{53} -3.01107i q^{55} +(3.64838 + 6.31917i) q^{59} +(-1.74194 - 1.00571i) q^{61} +(-1.80692 - 1.04322i) q^{65} +(1.20645 + 2.08964i) q^{67} +12.3890i q^{71} +(7.05561 + 4.07356i) q^{73} +(5.21789 - 4.71024i) q^{77} +(-2.58634 + 4.47967i) q^{79} +(7.56910 + 13.1101i) q^{83} +(-0.303134 + 0.525044i) q^{85} +(6.52948 + 11.3094i) q^{89} +(-1.01877 - 4.76312i) q^{91} +(-2.91464 + 1.68277i) q^{95} +(1.44234 - 0.832736i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 4 q^{7} - 12 q^{13} + 32 q^{25} - 24 q^{31} - 4 q^{37} - 4 q^{43} - 16 q^{49} - 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} + 12 q^{85} - 36 q^{91} - 12 q^{97}+O(q^{100}) $ 32 * q + 4 * q^7 - 12 * q^13 + 32 * q^25 - 24 * q^31 - 4 * q^37 - 4 * q^43 - 16 * q^49 - 12 * q^61 + 4 * q^67 - 36 * q^73 + 28 * q^79 + 12 * q^85 - 36 * q^91 - 12 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	−1.13332			−0.506835			−0.253417	−	0.967357i	$-0.581555\pi$
$5$	−1.13332			−0.506835			−0.253417	+	0.967357i	$0.581555\pi$
$6$		0			0
$7$	−1.77285	−	1.96392i	−0.670076	−	0.742293i
$7$	−1.77285	−	1.96392i	−0.670076	−	0.742293i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			2.65687i			0.801076i	0.916280	+	0.400538i	$0.131177\pi$
$11$			2.65687i			0.801076i	−0.916280	+	0.400538i	$0.868823\pi$
$12$		0			0
$13$	1.59436	+	0.920505i	0.442196	+	0.255302i	0.704529	−	0.709675i	$-0.251158\pi$
$13$	1.59436	+	0.920505i	0.442196	+	0.255302i	−0.262333	+	0.964978i	$0.584492\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.267475	−	0.463281i	0.0648723	−	0.112362i	−0.831765	−	0.555128i	$-0.812669\pi$
$17$	0.267475	−	0.463281i	0.0648723	−	0.112362i	0.896637	+	0.442766i	$0.146003\pi$
$18$		0			0
$19$	2.57178	−	1.48482i	0.590006	−	0.340640i	−0.175094	−	0.984552i	$-0.556023\pi$
$19$	2.57178	−	1.48482i	0.590006	−	0.340640i	0.765100	+	0.643912i	$0.222690\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			0.969928i			0.202244i	0.994874	+	0.101122i	$0.0322432\pi$
$23$			0.969928i			0.202244i	−0.994874	+	0.101122i	$0.967757\pi$
$24$		0			0
$25$	−3.71559			−0.743119
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.66955	+	1.54127i	−0.495724	+	0.286206i	−0.726946	−	0.686695i	$-0.759061\pi$
$29$	−2.66955	+	1.54127i	−0.495724	+	0.286206i	0.231222	+	0.972901i	$0.425728\pi$
$30$		0			0
$31$	−0.682710	+	0.394163i	−0.122618	+	0.0707937i	−0.560055	−	0.828456i	$-0.689220\pi$
$31$	−0.682710	+	0.394163i	−0.122618	+	0.0707937i	0.437436	+	0.899249i	$0.355887\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00920	+	2.22575i	0.339618	+	0.376220i
$36$		0			0
$37$	−1.73316	−	3.00191i	−0.284929	−	0.493511i	0.687663	−	0.726030i	$-0.258637\pi$
$37$	−1.73316	−	3.00191i	−0.284929	−	0.493511i	−0.972592	+	0.232519i	$0.925303\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.22822	−	3.85939i	0.347990	−	0.602736i	−0.637903	−	0.770117i	$-0.720198\pi$
$41$	2.22822	−	3.85939i	0.347990	−	0.602736i	0.985892	+	0.167381i	$0.0535311\pi$
$42$		0			0
$43$	−0.849465	−	1.47132i	−0.129542	−	0.224374i	0.793957	−	0.607974i	$-0.208017\pi$
$43$	−0.849465	−	1.47132i	−0.129542	−	0.224374i	−0.923499	+	0.383600i	$0.874684\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.14286	+	8.90769i	−0.750163	+	1.29932i	0.197580	+	0.980287i	$0.436692\pi$
$47$	−5.14286	+	8.90769i	−0.750163	+	1.29932i	−0.947743	+	0.319034i	$0.896642\pi$
$48$		0			0
$49$	−0.713980	+	6.96349i	−0.101997	+	0.994785i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	11.9002	+	6.87059i	1.63462	+	0.943749i	0.982642	+	0.185511i	$0.0593942\pi$
$53$	11.9002	+	6.87059i	1.63462	+	0.943749i	0.651979	+	0.758237i	$0.273939\pi$
$54$		0			0
$55$		−	3.01107i		−	0.406013i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.64838	+	6.31917i	0.474978	+	0.822686i	0.999589	−	0.0286558i	$-0.00912267\pi$
$59$	3.64838	+	6.31917i	0.474978	+	0.822686i	−0.524611	+	0.851342i	$0.675789\pi$
$60$		0			0
$61$	−1.74194	−	1.00571i	−0.223033	−	0.128768i	0.384321	−	0.923200i	$-0.374436\pi$
$61$	−1.74194	−	1.00571i	−0.223033	−	0.128768i	−0.607354	+	0.794432i	$0.707769\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.80692	−	1.04322i	−0.224120	−	0.129396i
$66$		0			0
$67$	1.20645	+	2.08964i	0.147392	+	0.255290i	0.930263	−	0.366894i	$-0.119579\pi$
$67$	1.20645	+	2.08964i	0.147392	+	0.255290i	−0.782871	+	0.622184i	$0.786246\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.3890i			1.47030i	0.677905	+	0.735150i	$0.262888\pi$
$71$			12.3890i			1.47030i	−0.677905	+	0.735150i	$0.737112\pi$
$72$		0			0
$73$	7.05561	+	4.07356i	0.825796	+	0.476774i	0.852411	−	0.522872i	$-0.175139\pi$
$73$	7.05561	+	4.07356i	0.825796	+	0.476774i	−0.0266149	+	0.999646i	$0.508473\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.21789	−	4.71024i	0.594633	−	0.536782i
$78$		0			0
$79$	−2.58634	+	4.47967i	−0.290986	+	0.504002i	−0.974043	−	0.226363i	$-0.927317\pi$
$79$	−2.58634	+	4.47967i	−0.290986	+	0.504002i	0.683057	+	0.730365i	$0.260650\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.56910	+	13.1101i	0.830817	+	1.43902i	0.897391	+	0.441235i	$0.145460\pi$
$83$	7.56910	+	13.1101i	0.830817	+	1.43902i	−0.0665746	+	0.997781i	$0.521207\pi$
$84$		0			0
$85$	−0.303134	+	0.525044i	−0.0328795	+	0.0569490i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.52948	+	11.3094i	0.692123	+	1.19879i	0.971141	+	0.238506i	$0.0766578\pi$
$89$	6.52948	+	11.3094i	0.692123	+	1.19879i	−0.279018	+	0.960286i	$0.590009\pi$
$90$		0			0
$91$	−1.01877	−	4.76312i	−0.106796	−	0.499311i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.91464	+	1.68277i	−0.299035	+	0.172648i
$96$		0			0
$97$	1.44234	−	0.832736i	0.146448	−	0.0845515i	−0.424986	−	0.905200i	$-0.639721\pi$
$97$	1.44234	−	0.832736i	0.146448	−	0.0845515i	0.571433	+	0.820649i	$0.306388\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.48905			−0.247670			−0.123835	−	0.992303i	$-0.539519\pi$
$101$	−2.48905			−0.247670			−0.123835	+	0.992303i	$0.539519\pi$
$102$		0			0
$103$			6.64802i			0.655049i	0.944843	+	0.327524i	$0.106214\pi$
$103$			6.64802i			0.655049i	−0.944843	+	0.327524i	$0.893786\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.217160	−	0.125377i	0.0209936	−	0.0121207i	−0.489466	−	0.872022i	$-0.662808\pi$
$107$	0.217160	−	0.125377i	0.0209936	−	0.0121207i	0.510460	+	0.859901i	$0.329475\pi$
$108$		0			0
$109$	1.27998	−	2.21699i	0.122600	−	0.212349i	−0.798192	−	0.602403i	$-0.794210\pi$
$109$	1.27998	−	2.21699i	0.122600	−	0.212349i	0.920792	+	0.390053i	$0.127543\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.383124	+	0.221197i	0.0360413	+	0.0208085i	0.517912	−	0.855434i	$-0.326709\pi$
$113$	0.383124	+	0.221197i	0.0360413	+	0.0208085i	−0.481871	+	0.876242i	$0.660043\pi$
$114$		0			0
$115$		−	1.09924i		−	0.102504i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.38404	+	0.296028i	−0.126875	+	0.0271369i
$120$		0			0
$121$	3.94104			0.358277
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.87753			0.883473
$126$		0			0
$127$	17.6161			1.56317			0.781586	−	0.623797i	$-0.214411\pi$
$127$	17.6161			1.56317			0.781586	+	0.623797i	$0.214411\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.1985			−0.891049			−0.445525	−	0.895270i	$-0.646983\pi$
$131$	−10.1985			−0.891049			−0.445525	+	0.895270i	$0.646983\pi$
$132$		0			0
$133$	−7.47545	−	2.41841i	−0.648203	−	0.209703i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.41103i			0.718603i	0.933222	+	0.359301i	$0.116985\pi$
$137$			8.41103i			0.718603i	−0.933222	+	0.359301i	$0.883015\pi$
$138$		0			0
$139$	8.83742	+	5.10229i	0.749580	+	0.432770i	0.825542	−	0.564340i	$-0.190869\pi$
$139$	8.83742	+	5.10229i	0.749580	+	0.432770i	−0.0759620	+	0.997111i	$0.524203\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.44566	+	4.23601i	−0.204517	+	0.354233i
$144$		0			0
$145$	3.02545	−	1.74674i	0.251250	−	0.145059i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	15.1997i		−	1.24521i	−0.782536	−	0.622605i	$-0.786074\pi$
$149$		−	15.1997i		−	1.24521i	0.782536	−	0.622605i	$-0.213926\pi$
$150$		0			0
$151$	−8.40245			−0.683781			−0.341891	−	0.939740i	$-0.611067\pi$
$151$	−8.40245			−0.683781			−0.341891	+	0.939740i	$0.611067\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.773727	−	0.446711i	0.0621472	−	0.0358807i
$156$		0			0
$157$	−16.1793	+	9.34111i	−1.29125	+	0.745501i	−0.978875	−	0.204459i	$-0.934457\pi$
$157$	−16.1793	+	9.34111i	−1.29125	+	0.745501i	−0.312371	+	0.949960i	$0.601123\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.90486	−	1.71954i	0.150124	−	0.135519i
$162$		0			0
$163$	−10.7857	−	18.6813i	−0.844800	−	1.46324i	−0.885795	−	0.464077i	$-0.846386\pi$
$163$	−10.7857	−	18.6813i	−0.844800	−	1.46324i	0.0409955	−	0.999159i	$-0.486947\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.39810	+	11.0818i	−0.495099	+	0.857537i	−0.999984	−	0.00564941i	$-0.998202\pi$
$167$	−6.39810	+	11.0818i	−0.495099	+	0.857537i	0.504885	+	0.863187i	$0.331535\pi$
$168$		0			0
$169$	−4.80534	−	8.32310i	−0.369642	−	0.640238i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.6999	+	18.5328i	−0.813501	+	1.40903i	0.0968975	+	0.995294i	$0.469108\pi$
$173$	−10.6999	+	18.5328i	−0.813501	+	1.40903i	−0.910399	+	0.413732i	$0.864225\pi$
$174$		0			0
$175$	6.58720	+	7.29714i	0.497946	+	0.551612i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.36953	−	2.52275i	−0.326594	−	0.188559i	0.327734	−	0.944770i	$-0.393715\pi$
$179$	−4.36953	−	2.52275i	−0.326594	−	0.188559i	−0.654328	+	0.756211i	$0.727048\pi$
$180$		0			0
$181$			11.6959i			0.869351i	0.900587	+	0.434675i	$0.143137\pi$
$181$			11.6959i			0.869351i	−0.900587	+	0.434675i	$0.856863\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.96421	+	3.40212i	0.144412	+	0.250129i
$186$		0			0
$187$	1.23088	+	0.710647i	0.0900106	+	0.0519676i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.6083	−	11.3209i	−1.41881	−	0.819148i	−0.422612	−	0.906311i	$-0.638887\pi$
$191$	−19.6083	−	11.3209i	−1.41881	−	0.819148i	−0.996194	+	0.0871622i	$0.972220\pi$
$192$		0			0
$193$	0.559276	+	0.968695i	0.0402576	+	0.0697282i	0.885452	−	0.464731i	$-0.153849\pi$
$193$	0.559276	+	0.968695i	0.0402576	+	0.0697282i	−0.845195	+	0.534459i	$0.820515\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3.19138i		−	0.227377i	−0.993516	−	0.113688i	$-0.963733\pi$
$197$		−	3.19138i		−	0.227377i	0.993516	−	0.113688i	$-0.0362665\pi$
$198$		0			0
$199$	0.346408	+	0.199999i	0.0245562	+	0.0141775i	0.512228	−	0.858850i	$-0.328820\pi$
$199$	0.346408	+	0.199999i	0.0245562	+	0.0141775i	−0.487672	+	0.873027i	$0.662154\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.75966	+	2.51035i	0.544621	+	0.176192i
$204$		0			0
$205$	−2.52528	+	4.37391i	−0.176373	+	0.305487i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.94496	+	6.83288i	0.272879	+	0.472640i
$210$		0			0
$211$	−2.82379	+	4.89095i	−0.194398	+	0.336707i	−0.946703	−	0.322108i	$-0.895609\pi$
$211$	−2.82379	+	4.89095i	−0.194398	+	0.336707i	0.752305	+	0.658815i	$0.228942\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.962712	+	1.66747i	0.0656564	+	0.113720i
$216$		0			0
$217$	1.98445	+	0.641996i	0.134713	+	0.0435816i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.852904	−	0.492425i	0.0573725	−	0.0331241i
$222$		0			0
$223$	−5.40211	+	3.11891i	−0.361752	+	0.208858i	−0.669849	−	0.742497i	$-0.733641\pi$
$223$	−5.40211	+	3.11891i	−0.361752	+	0.208858i	0.308097	+	0.951355i	$0.400308\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.3288			1.08378			0.541892	−	0.840448i	$-0.317708\pi$
$227$	16.3288			1.08378			0.541892	+	0.840448i	$0.317708\pi$
$228$		0			0
$229$		−	23.4203i		−	1.54765i	−0.633397	−	0.773827i	$-0.718340\pi$
$229$		−	23.4203i		−	1.54765i	0.633397	−	0.773827i	$-0.281660\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.9345	−	6.31304i	0.716343	−	0.413581i	−0.0970623	−	0.995278i	$-0.530945\pi$
$233$	10.9345	−	6.31304i	0.716343	−	0.413581i	0.813405	+	0.581698i	$0.197611\pi$
$234$		0			0
$235$	5.82849	−	10.0952i	0.380209	−	0.658541i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.56517	+	4.94510i	0.554035	+	0.319872i	0.750748	−	0.660589i	$-0.229693\pi$
$239$	8.56517	+	4.94510i	0.554035	+	0.319872i	−0.196713	+	0.980461i	$0.563027\pi$
$240$		0			0
$241$		−	20.2198i		−	1.30247i	−0.758875	−	0.651236i	$-0.774251\pi$
$241$		−	20.2198i		−	1.30247i	0.758875	−	0.651236i	$-0.225749\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.809165	−	7.89184i	0.0516957	−	0.504191i
$246$		0			0
$247$	5.46712			0.347865
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10.4354			−0.658679			−0.329339	−	0.944212i	$-0.606826\pi$
$251$	−10.4354			−0.658679			−0.329339	+	0.944212i	$0.606826\pi$
$252$		0			0
$253$	−2.57697			−0.162013
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.81677			−0.487597			−0.243798	−	0.969826i	$-0.578394\pi$
$257$	−7.81677			−0.487597			−0.243798	+	0.969826i	$0.578394\pi$
$258$		0			0
$259$	−2.82289	+	8.72573i	−0.175406	+	0.542191i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			1.70199i			0.104949i	0.998622	+	0.0524745i	$0.0167108\pi$
$263$			1.70199i			0.104949i	−0.998622	+	0.0524745i	$0.983289\pi$
$264$		0			0
$265$	−13.4867	−	7.78656i	−0.828482	−	0.478324i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.59447	−	4.49375i	0.158188	−	0.273989i	−0.776028	−	0.630699i	$-0.782768\pi$
$269$	2.59447	−	4.49375i	0.158188	−	0.273989i	0.934215	+	0.356710i	$0.116102\pi$
$270$		0			0
$271$	−22.7985	+	13.1627i	−1.38491	+	0.799579i	−0.992736	−	0.120311i	$-0.961611\pi$
$271$	−22.7985	+	13.1627i	−1.38491	+	0.799579i	−0.392176	+	0.919890i	$0.628278\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	9.87185i		−	0.595295i
$276$		0			0
$277$	23.5952			1.41770			0.708849	−	0.705360i	$-0.249215\pi$
$277$	23.5952			1.41770			0.708849	+	0.705360i	$0.249215\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.0083	−	9.24241i	0.954976	−	0.551356i	0.0603530	−	0.998177i	$-0.480777\pi$
$281$	16.0083	−	9.24241i	0.954976	−	0.551356i	0.894623	+	0.446821i	$0.147444\pi$
$282$		0			0
$283$	15.3990	−	8.89059i	0.915373	−	0.528491i	0.0332168	−	0.999448i	$-0.489425\pi$
$283$	15.3990	−	8.89059i	0.915373	−	0.528491i	0.882156	+	0.470958i	$0.156091\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−11.5299	+	2.46609i	−0.680586	+	0.145568i
$288$		0			0
$289$	8.35691	+	14.4746i	0.491583	+	0.851447i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.43356	−	4.21505i	0.142170	−	0.246246i	−0.786143	−	0.618044i	$-0.787925\pi$
$293$	2.43356	−	4.21505i	0.142170	−	0.246246i	0.928314	+	0.371798i	$0.121259\pi$
$294$		0			0
$295$	−4.13476	−	7.16162i	−0.240735	−	0.416966i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.892823	+	1.54642i	−0.0516333	+	0.0894315i
$300$		0			0
$301$	−1.38357	+	4.27671i	−0.0797479	+	0.246505i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.97417	+	1.13979i	0.113041	+	0.0652641i
$306$		0			0
$307$			8.37189i			0.477809i	0.971043	+	0.238905i	$0.0767883\pi$
$307$			8.37189i			0.477809i	−0.971043	+	0.238905i	$0.923212\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.10547	+	14.0391i	0.459619	+	0.796084i	0.998941	−	0.0460162i	$-0.0146526\pi$
$311$	8.10547	+	14.0391i	0.459619	+	0.796084i	−0.539322	+	0.842100i	$0.681319\pi$
$312$		0			0
$313$	19.9011	+	11.4899i	1.12488	+	0.649449i	0.942642	−	0.333806i	$-0.108333\pi$
$313$	19.9011	+	11.4899i	1.12488	+	0.649449i	0.182237	+	0.983255i	$0.441666\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−18.8322	−	10.8728i	−1.05772	−	0.610675i	−0.132920	−	0.991127i	$-0.542435\pi$
$317$	−18.8322	−	10.8728i	−1.05772	−	0.610675i	−0.924801	+	0.380452i	$0.875769\pi$
$318$		0			0
$319$	−4.09495	−	7.09266i	−0.229273	−	0.397113i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1.58861i		−	0.0883924i
$324$		0			0
$325$	−5.92400	−	3.42022i	−0.328604	−	0.189720i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	26.6116	−	5.69186i	1.46714	−	0.313802i
$330$		0			0
$331$	−12.8868	+	22.3207i	−0.708325	+	1.22685i	0.257154	+	0.966371i	$0.417215\pi$
$331$	−12.8868	+	22.3207i	−0.708325	+	1.22685i	−0.965478	+	0.260484i	$0.916118\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.36729	−	2.36822i	−0.0747032	−	0.129390i
$336$		0			0
$337$	6.80663	−	11.7894i	0.370781	−	0.642211i	−0.618905	−	0.785466i	$-0.712423\pi$
$337$	6.80663	−	11.7894i	0.370781	−	0.642211i	0.989686	+	0.143255i	$0.0457568\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.04724	−	1.81387i	−0.0567112	−	0.0982267i
$342$		0			0
$343$	14.9415	−	10.9431i	0.806767	−	0.590869i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.35635	+	4.82454i	−0.448593	+	0.258995i	−0.707236	−	0.706978i	$-0.750058\pi$
$347$	−8.35635	+	4.82454i	−0.448593	+	0.258995i	0.258643	+	0.965973i	$0.416725\pi$
$348$		0			0
$349$	20.1890	−	11.6561i	1.08069	−	0.623937i	0.149608	−	0.988745i	$-0.452199\pi$
$349$	20.1890	−	11.6561i	1.08069	−	0.623937i	0.931083	+	0.364808i	$0.118865\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	31.0020			1.65007			0.825035	−	0.565081i	$-0.191155\pi$
$353$	31.0020			1.65007			0.825035	+	0.565081i	$0.191155\pi$
$354$		0			0
$355$		−	14.0406i		−	0.745198i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.6898	+	9.05853i	−0.828077	+	0.478091i	−0.853194	−	0.521594i	$-0.825338\pi$
$359$	−15.6898	+	9.05853i	−0.828077	+	0.478091i	0.0251166	+	0.999685i	$0.492004\pi$
$360$		0			0
$361$	−5.09064	+	8.81725i	−0.267929	+	0.464066i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.99624	−	4.61663i	−0.418542	−	0.241645i
$366$		0			0
$367$		−	3.16219i		−	0.165065i	−0.996588	−	0.0825326i	$-0.973699\pi$
$367$		−	3.16219i		−	0.165065i	0.996588	−	0.0825326i	$-0.0263008\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.60403	−	35.5517i	−0.394782	−	1.84575i
$372$		0			0
$373$	7.37186			0.381700			0.190850	−	0.981619i	$-0.438876\pi$
$373$	7.37186			0.381700			0.190850	+	0.981619i	$0.438876\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.67498			−0.292276
$378$		0			0
$379$	−17.7598			−0.912259			−0.456130	−	0.889913i	$-0.650765\pi$
$379$	−17.7598			−0.912259			−0.456130	+	0.889913i	$0.650765\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−21.2701			−1.08685			−0.543425	−	0.839458i	$-0.682873\pi$
$383$	−21.2701			−1.08685			−0.543425	+	0.839458i	$0.682873\pi$
$384$		0			0
$385$	−5.91352	+	5.33819i	−0.301381	+	0.272060i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	3.67016i		−	0.186085i	−0.995662	−	0.0930424i	$-0.970341\pi$
$389$		−	3.67016i		−	0.186085i	0.995662	−	0.0930424i	$-0.0296592\pi$
$390$		0			0
$391$	0.449349	+	0.259432i	0.0227245	+	0.0131200i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.93114	−	5.07688i	0.147482	−	0.255446i
$396$		0			0
$397$	−16.4664	+	9.50687i	−0.826424	+	0.477136i	−0.852627	−	0.522521i	$-0.824992\pi$
$397$	−16.4664	+	9.50687i	−0.826424	+	0.477136i	0.0262027	+	0.999657i	$0.491658\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	31.0462i		−	1.55037i	−0.631733	−	0.775186i	$-0.717656\pi$
$401$		−	31.0462i		−	1.55037i	0.631733	−	0.775186i	$-0.282344\pi$
$402$		0			0
$403$	−1.45132			−0.0722952
$404$		0			0
$405$		0			0
$406$		0			0
$407$	7.97569	−	4.60477i	0.395340	−	0.228250i
$408$		0			0
$409$	−30.4429	+	17.5762i	−1.50530	+	0.869087i	−0.505322	+	0.862931i	$0.668626\pi$
$409$	−30.4429	+	17.5762i	−1.50530	+	0.869087i	−0.999981	+	0.00615678i	$0.998040\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.94233	−	18.3681i	0.292403	−	0.903835i
$414$		0			0
$415$	−8.57819	−	14.8579i	−0.421087	−	0.729344i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−14.9516	+	25.8970i	−0.730436	+	1.26515i	0.226261	+	0.974067i	$0.427350\pi$
$419$	−14.9516	+	25.8970i	−0.730436	+	1.26515i	−0.956697	+	0.291085i	$0.905984\pi$
$420$		0			0
$421$	13.0842	+	22.6624i	0.637683	+	1.10450i	0.985940	+	0.167100i	$0.0534402\pi$
$421$	13.0842	+	22.6624i	0.637683	+	1.10450i	−0.348257	+	0.937399i	$0.613226\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.993829	+	1.72136i	−0.0482078	+	0.0834984i
$426$		0			0
$427$	1.11307	+	5.20401i	0.0538652	+	0.251840i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−34.8031	−	20.0936i	−1.67641	−	0.967875i	−0.963921	−	0.266189i	$-0.914235\pi$
$431$	−34.8031	−	20.0936i	−1.67641	−	0.967875i	−0.712487	−	0.701685i	$-0.752431\pi$
$432$		0			0
$433$			15.4379i			0.741896i	0.928654	+	0.370948i	$0.120967\pi$
$433$			15.4379i			0.741896i	−0.928654	+	0.370948i	$0.879033\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.44016	+	2.49444i	0.0688924	+	0.119325i
$438$		0			0
$439$	21.6747	+	12.5139i	1.03447	+	0.597254i	0.918263	−	0.395970i	$-0.129592\pi$
$439$	21.6747	+	12.5139i	1.03447	+	0.597254i	0.116212	+	0.993224i	$0.462925\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.6906	−	7.32694i	−0.602950	−	0.348113i	0.167251	−	0.985914i	$-0.446511\pi$
$443$	−12.6906	−	7.32694i	−0.602950	−	0.348113i	−0.770201	+	0.637801i	$0.779844\pi$
$444$		0			0
$445$	−7.39996	−	12.8171i	−0.350792	−	0.607589i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	31.1359i		−	1.46939i	−0.678395	−	0.734697i	$-0.737324\pi$
$449$		−	31.1359i		−	1.46939i	0.678395	−	0.734697i	$-0.262676\pi$
$450$		0			0
$451$	10.2539	+	5.92009i	0.482838	+	0.278766i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.15459	+	5.39812i	0.0541279	+	0.253068i
$456$		0			0
$457$	10.8976	−	18.8752i	0.509769	−	0.882945i	−0.490167	−	0.871628i	$-0.663064\pi$
$457$	10.8976	−	18.8752i	0.509769	−	0.882945i	0.999936	−	0.0113168i	$-0.00360234\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−13.3754	−	23.1670i	−0.622957	−	1.07899i	−0.988932	−	0.148368i	$-0.952598\pi$
$461$	−13.3754	−	23.1670i	−0.622957	−	1.07899i	0.365976	−	0.930624i	$-0.380735\pi$
$462$		0			0
$463$	−7.62549	+	13.2077i	−0.354386	+	0.613815i	−0.987013	−	0.160642i	$-0.948644\pi$
$463$	−7.62549	+	13.2077i	−0.354386	+	0.613815i	0.632626	+	0.774457i	$0.281977\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−17.9707	−	31.1262i	−0.831585	−	1.44035i	−0.896781	−	0.442475i	$-0.854101\pi$
$467$	−17.9707	−	31.1262i	−0.831585	−	1.44035i	0.0651959	−	0.997872i	$-0.479233\pi$
$468$		0			0
$469$	1.96502	−	6.07400i	0.0907363	−	0.280471i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.90910	−	2.25692i	0.179740	−	0.103773i
$474$		0			0
$475$	−9.55568	+	5.51697i	−0.438445	+	0.253136i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	30.9451			1.41392			0.706959	−	0.707254i	$-0.250066\pi$
$479$	30.9451			1.41392			0.706959	+	0.707254i	$0.250066\pi$
$480$		0			0
$481$		−	6.38151i		−	0.290972i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.63463	+	0.943753i	−0.0742247	+	0.0428536i
$486$		0			0
$487$	−7.67907	+	13.3005i	−0.347972	+	0.602705i	−0.985889	−	0.167400i	$-0.946463\pi$
$487$	−7.67907	+	13.3005i	−0.347972	+	0.602705i	0.637917	+	0.770105i	$0.279796\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.105233	+	0.0607560i	0.00474908	+	0.00274188i	0.502373	−	0.864651i	$-0.332461\pi$
$491$	0.105233	+	0.0607560i	0.00474908	+	0.00274188i	−0.497624	+	0.867393i	$0.665794\pi$
$492$		0			0
$493$			1.64900i			0.0742674i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	24.3310	−	21.9638i	1.09139	−	0.985212i
$498$		0			0
$499$	25.6678			1.14905			0.574524	−	0.818488i	$-0.305187\pi$
$499$	25.6678			1.14905			0.574524	+	0.818488i	$0.305187\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.44680			−0.153685			−0.0768425	−	0.997043i	$-0.524484\pi$
$503$	−3.44680			−0.153685			−0.0768425	+	0.997043i	$0.524484\pi$
$504$		0			0
$505$	2.82088			0.125528
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−40.2989			−1.78622			−0.893109	−	0.449840i	$-0.851481\pi$
$509$	−40.2989			−1.78622			−0.893109	+	0.449840i	$0.851481\pi$
$510$		0			0
$511$	−4.50841	−	21.0785i	−0.199440	−	0.932457i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	7.53431i		−	0.332001i
$516$		0			0
$517$	−23.6666	−	13.6639i	−1.04086	−	0.600938i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.12248	−	3.67624i	0.0929874	−	0.161059i	−0.815779	−	0.578363i	$-0.803692\pi$
$521$	2.12248	−	3.67624i	0.0929874	−	0.161059i	0.908767	+	0.417304i	$0.137025\pi$
$522$		0			0
$523$	−15.9490	+	9.20818i	−0.697403	+	0.402646i	−0.806379	−	0.591399i	$-0.798576\pi$
$523$	−15.9490	+	9.20818i	−0.697403	+	0.402646i	0.108977	+	0.994044i	$0.465243\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			0.421715i			0.0183702i
$528$		0			0
$529$	22.0592			0.959097
$530$		0			0
$531$		0			0
$532$		0			0
$533$	7.10518	−	4.10218i	0.307760	−	0.177685i
$534$		0			0
$535$	−0.246111	+	0.142092i	−0.0106403	+	0.00614318i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−18.5011	−	1.89695i	−0.796899	−	0.0817075i
$540$		0			0
$541$	−11.7817	−	20.4065i	−0.506535	−	0.877344i	−0.999971	−	0.00756235i	$-0.997593\pi$
$541$	−11.7817	−	20.4065i	−0.506535	−	0.877344i	0.493437	−	0.869782i	$-0.335741\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.45062	+	2.51256i	−0.0621379	+	0.107626i
$546$		0			0
$547$	4.87688	+	8.44701i	0.208521	+	0.361168i	0.951249	−	0.308425i	$-0.0998019\pi$
$547$	4.87688	+	8.44701i	0.208521	+	0.361168i	−0.742728	+	0.669593i	$0.766469\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.57700	+	7.92759i	−0.194987	+	0.337727i
$552$		0			0
$553$	13.3829	−	2.86243i	0.569100	−	0.121723i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.22763	−	4.75023i	−0.348616	−	0.201274i	0.315460	−	0.948939i	$-0.397841\pi$
$557$	−8.22763	−	4.75023i	−0.348616	−	0.201274i	−0.664076	+	0.747666i	$0.731175\pi$
$558$		0			0
$559$		−	3.12775i		−	0.132290i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.1895	+	24.5769i	0.598014	+	1.03579i	0.993114	+	0.117153i	$0.0373768\pi$
$563$	14.1895	+	24.5769i	0.598014	+	1.03579i	−0.395100	+	0.918638i	$0.629290\pi$
$564$		0			0
$565$	−0.434201	−	0.250686i	−0.0182670	−	0.0105464i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−19.8968	−	11.4874i	−0.834118	−	0.481578i	0.0211427	−	0.999776i	$-0.493270\pi$
$569$	−19.8968	−	11.4874i	−0.834118	−	0.481578i	−0.855260	+	0.518198i	$0.826603\pi$
$570$		0			0
$571$	−4.50245	−	7.79848i	−0.188422	−	0.326356i	0.756302	−	0.654222i	$-0.227004\pi$
$571$	−4.50245	−	7.79848i	−0.188422	−	0.326356i	−0.944724	+	0.327866i	$0.893671\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	3.60386i		−	0.150291i
$576$		0			0
$577$	31.7024	+	18.3034i	1.31979	+	0.761980i	0.983694	−	0.179848i	$-0.0575607\pi$
$577$	31.7024	+	18.3034i	1.31979	+	0.761980i	0.336094	+	0.941828i	$0.390894\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12.3282	−	38.1074i	0.511462	−	1.58096i
$582$		0			0
$583$	−18.2543	+	31.6173i	−0.756015	+	1.30946i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.3397	+	23.1051i	0.550589	+	0.953649i	0.998232	+	0.0594361i	$0.0189303\pi$
$587$	13.3397	+	23.1051i	0.550589	+	0.953649i	−0.447643	+	0.894212i	$0.647736\pi$
$588$		0			0
$589$	−1.17052	+	2.02740i	−0.0482304	+	0.0835375i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.167630	+	0.290343i	0.00688372	+	0.0119230i	0.869447	−	0.494027i	$-0.164475\pi$
$593$	0.167630	+	0.290343i	0.00688372	+	0.0119230i	−0.862563	+	0.505950i	$0.831142\pi$
$594$		0			0
$595$	1.56856	−	0.335494i	0.0643046	−	0.0137539i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−14.7019	+	8.48812i	−0.600702	+	0.346815i	−0.769318	−	0.638867i	$-0.779404\pi$
$599$	−14.7019	+	8.48812i	−0.600702	+	0.346815i	0.168616	+	0.985682i	$0.446070\pi$
$600$		0			0
$601$	14.5160	−	8.38082i	0.592120	−	0.341861i	−0.173815	−	0.984778i	$-0.555609\pi$
$601$	14.5160	−	8.38082i	0.592120	−	0.341861i	0.765936	+	0.642917i	$0.222276\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.46645			−0.181587
$606$		0			0
$607$		−	41.6356i		−	1.68994i	−0.534816	−	0.844968i	$-0.679619\pi$
$607$		−	41.6356i		−	1.68994i	0.534816	−	0.844968i	$-0.320381\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−16.3991	+	9.46805i	−0.663439	+	0.383036i
$612$		0			0
$613$	−12.8608	+	22.2756i	−0.519444	+	0.899704i	0.480300	+	0.877104i	$0.340528\pi$
$613$	−12.8608	+	22.2756i	−0.519444	+	0.899704i	−0.999745	+	0.0225996i	$0.992806\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	37.0394	+	21.3847i	1.49115	+	0.860915i	0.999949	−	0.0101324i	$-0.00322529\pi$
$617$	37.0394	+	21.3847i	1.49115	+	0.860915i	0.491199	+	0.871047i	$0.336559\pi$
$618$		0			0
$619$		−	27.6519i		−	1.11142i	−0.831375	−	0.555712i	$-0.812446\pi$
$619$		−	27.6519i		−	1.11142i	0.831375	−	0.555712i	$-0.187554\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10.6349	−	32.8733i	0.426080	−	1.31704i
$624$		0			0
$625$	7.38360			0.295344
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.85430			−0.0739359
$630$		0			0
$631$	2.23244			0.0888720			0.0444360	−	0.999012i	$-0.485851\pi$
$631$	2.23244			0.0888720			0.0444360	+	0.999012i	$0.485851\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−19.9646			−0.792270
$636$		0			0
$637$	−7.54827	+	10.4451i	−0.299073	+	0.413850i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	27.0409i		−	1.06805i	−0.845468	−	0.534026i	$-0.820679\pi$
$641$		−	27.0409i		−	1.06805i	0.845468	−	0.534026i	$-0.179321\pi$
$642$		0			0
$643$	−10.4874	−	6.05488i	−0.413581	−	0.238781i	0.278746	−	0.960365i	$-0.410081\pi$
$643$	−10.4874	−	6.05488i	−0.413581	−	0.238781i	−0.692327	+	0.721584i	$0.743415\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.40271	−	5.89366i	0.133774	−	0.231704i	−0.791354	−	0.611358i	$-0.790624\pi$
$647$	3.40271	−	5.89366i	0.133774	−	0.231704i	0.925129	+	0.379654i	$0.123957\pi$
$648$		0			0
$649$	−16.7892	+	9.69326i	−0.659034	+	0.380494i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	45.9211i		−	1.79703i	−0.438940	−	0.898516i	$-0.644646\pi$
$653$		−	45.9211i		−	1.79703i	0.438940	−	0.898516i	$-0.355354\pi$
$654$		0			0
$655$	11.5582			0.451615
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.9825	−	7.49548i	0.505728	−	0.291982i	−0.225348	−	0.974278i	$-0.572352\pi$
$659$	12.9825	−	7.49548i	0.505728	−	0.291982i	0.731076	+	0.682296i	$0.239018\pi$
$660$		0			0
$661$	28.6619	−	16.5479i	1.11482	−	0.643640i	0.174745	−	0.984614i	$-0.444090\pi$
$661$	28.6619	−	16.5479i	1.11482	−	0.643640i	0.940073	+	0.340974i	$0.110757\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.47205	+	2.74082i	0.328532	+	0.106285i
$666$		0			0
$667$	−1.49492	−	2.58927i	−0.0578835	−	0.100257i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.67204	−	4.62811i	0.103153	−	0.178666i
$672$		0			0
$673$	−18.6158	−	32.2436i	−0.717588	−	1.24290i	−0.961953	−	0.273215i	$-0.911913\pi$
$673$	−18.6158	−	32.2436i	−0.717588	−	1.24290i	0.244365	−	0.969683i	$-0.421420\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−20.1462	+	34.8942i	−0.774280	+	1.34109i	0.160919	+	0.986968i	$0.448554\pi$
$677$	−20.1462	+	34.8942i	−0.774280	+	1.34109i	−0.935198	+	0.354124i	$0.884779\pi$
$678$		0			0
$679$	−4.19249	−	1.35633i	−0.160893	−	0.0520510i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.2066	+	7.04746i	0.467071	+	0.269664i	0.715013	−	0.699111i	$-0.246421\pi$
$683$	12.2066	+	7.04746i	0.467071	+	0.269664i	−0.247942	+	0.968775i	$0.579754\pi$
$684$		0			0
$685$		−	9.53236i		−	0.364213i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12.6488	+	21.9084i	0.481882	+	0.834644i
$690$		0			0
$691$	32.6993	+	18.8789i	1.24394	+	0.718188i	0.969894	−	0.243528i	$-0.0783049\pi$
$691$	32.6993	+	18.8789i	1.24394	+	0.718188i	0.274045	+	0.961717i	$0.411638\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.0156	−	5.78251i	−0.379913	−	0.219343i
$696$		0			0
$697$	−1.19199	−	2.06458i	−0.0451498	−	0.0782017i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	13.0860i		−	0.494252i	−0.968983	−	0.247126i	$-0.920514\pi$
$701$		−	13.0860i		−	0.494252i	0.968983	−	0.247126i	$-0.0794861\pi$
$702$		0			0
$703$	−8.91458	−	5.14683i	−0.336220	−	0.194116i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.41272	+	4.88830i	0.165957	+	0.183843i
$708$		0			0
$709$	−11.2777	+	19.5336i	−0.423544	+	0.733600i	−0.996283	−	0.0861375i	$-0.972548\pi$
$709$	−11.2777	+	19.5336i	−0.423544	+	0.733600i	0.572739	+	0.819738i	$0.305881\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.382310	−	0.662180i	−0.0143176	−	0.0247988i
$714$		0			0
$715$	2.77171	−	4.80074i	0.103656	−	0.179538i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−5.27786	−	9.14152i	−0.196831	−	0.340921i	0.750668	−	0.660679i	$-0.229732\pi$
$719$	−5.27786	−	9.14152i	−0.196831	−	0.340921i	−0.947499	+	0.319758i	$0.896398\pi$
$720$		0			0
$721$	13.0562	−	11.7860i	0.486238	−	0.438932i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.91898	−	5.72672i	0.368382	−	0.212685i
$726$		0			0
$727$	21.9441	−	12.6694i	0.813861	−	0.469883i	−0.0344341	−	0.999407i	$-0.510963\pi$
$727$	21.9441	−	12.6694i	0.813861	−	0.469883i	0.848295	+	0.529524i	$0.177630\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.908843			−0.0336148
$732$		0			0
$733$			3.82697i			0.141352i	0.997499	+	0.0706762i	$0.0225157\pi$
$733$			3.82697i			0.141352i	−0.997499	+	0.0706762i	$0.977484\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.55190	+	3.20539i	−0.204507	+	0.118072i
$738$		0			0
$739$	−18.1289	+	31.4003i	−0.666884	+	1.15508i	0.311887	+	0.950119i	$0.399039\pi$
$739$	−18.1289	+	31.4003i	−0.666884	+	1.15508i	−0.978771	+	0.204958i	$0.934294\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.3598	+	7.13594i	0.453438	+	0.261792i	0.709281	−	0.704926i	$-0.249020\pi$
$743$	12.3598	+	7.13594i	0.453438	+	0.261792i	−0.255843	+	0.966718i	$0.582353\pi$
$744$		0			0
$745$			17.2261i			0.631115i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.631224	−	0.204209i	−0.0230644	−	0.00746165i
$750$		0			0
$751$	29.2009			1.06556			0.532778	−	0.846255i	$-0.321148\pi$
$751$	29.2009			1.06556			0.532778	+	0.846255i	$0.321148\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9.52263			0.346564
$756$		0			0
$757$	−33.2051			−1.20686			−0.603431	−	0.797415i	$-0.706200\pi$
$757$	−33.2051			−1.20686			−0.603431	+	0.797415i	$0.706200\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	14.9369			0.541463			0.270731	−	0.962655i	$-0.412734\pi$
$761$	14.9369			0.541463			0.270731	+	0.962655i	$0.412734\pi$
$762$		0			0
$763$	−6.62322	+	1.41662i	−0.239777	+	0.0512851i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			13.4334i			0.485052i
$768$		0			0
$769$	−27.1703	−	15.6868i	−0.979788	−	0.565681i	−0.0775816	−	0.996986i	$-0.524720\pi$
$769$	−27.1703	−	15.6868i	−0.979788	−	0.565681i	−0.902206	+	0.431305i	$0.858053\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.464799	−	0.805055i	0.0167176	−	0.0289558i	−0.857546	−	0.514408i	$-0.828012\pi$
$773$	0.464799	−	0.805055i	0.0167176	−	0.0289558i	0.874263	+	0.485452i	$0.161345\pi$
$774$		0			0
$775$	2.53667	−	1.46455i	0.0911200	−	0.0526082i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	13.2340i		−	0.474157i
$780$		0			0
$781$	−32.9159			−1.17782
$782$		0			0
$783$		0			0
$784$		0			0
$785$	18.3362	−	10.5864i	0.654448	−	0.377846i
$786$		0			0
$787$	−34.9455	+	20.1758i	−1.24567	+	0.719189i	−0.970243	−	0.242132i	$-0.922153\pi$
$787$	−34.9455	+	20.1758i	−1.24567	+	0.719189i	−0.275429	+	0.961321i	$0.588820\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−0.244810	−	1.14458i	−0.00870443	−	0.0406964i
$792$		0			0
$793$	−1.85152	−	3.20693i	−0.0657495	−	0.113881i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−11.2121	+	19.4199i	−0.397152	+	0.687888i	−0.993373	−	0.114932i	$-0.963335\pi$
$797$	−11.2121	+	19.4199i	−0.397152	+	0.687888i	0.596221	+	0.802820i	$0.296668\pi$
$798$		0			0
$799$	2.75117	+	4.76517i	0.0973296	+	0.168580i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−10.8229	+	18.7458i	−0.381932	+	0.661526i
$804$		0			0
$805$	−2.15881	+	1.94878i	−0.0760881	+	0.0686856i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18.2271	+	10.5234i	0.640832	+	0.369985i	0.784935	−	0.619578i	$-0.212696\pi$
$809$	18.2271	+	10.5234i	0.640832	+	0.369985i	−0.144103	+	0.989563i	$0.546030\pi$
$810$		0			0
$811$		−	31.5558i		−	1.10808i	−0.832492	−	0.554038i	$-0.813086\pi$
$811$		−	31.5558i		−	1.10808i	0.832492	−	0.554038i	$-0.186914\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.2236	+	21.1719i	0.428174	+	0.741619i
$816$		0			0
$817$	−4.36927	−	2.52260i	−0.152861	−	0.0882545i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−47.3134	−	27.3164i	−1.65125	−	0.953349i	−0.976560	−	0.215246i	$-0.930945\pi$
$821$	−47.3134	−	27.3164i	−1.65125	−	0.953349i	−0.674688	−	0.738103i	$-0.735722\pi$
$822$		0			0
$823$	5.88746	+	10.1974i	0.205224	+	0.355458i	0.950204	−	0.311628i	$-0.100874\pi$
$823$	5.88746	+	10.1974i	0.205224	+	0.355458i	−0.744980	+	0.667087i	$0.767541\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			27.9059i			0.970383i	0.874408	+	0.485192i	$0.161250\pi$
$827$			27.9059i			0.970383i	−0.874408	+	0.485192i	$0.838750\pi$
$828$		0			0
$829$	21.7374	+	12.5501i	0.754972	+	0.435883i	0.827488	−	0.561484i	$-0.189769\pi$
$829$	21.7374	+	12.5501i	0.754972	+	0.435883i	−0.0725156	+	0.997367i	$0.523103\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.03508	+	2.19333i	0.105159	+	0.0759946i
$834$		0			0
$835$	7.25107	−	12.5592i	0.250934	−	0.434630i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	22.6654	+	39.2576i	0.782497	+	1.35532i	0.930483	+	0.366335i	$0.119387\pi$
$839$	22.6654	+	39.2576i	0.782497	+	1.35532i	−0.147987	+	0.988989i	$0.547279\pi$
$840$		0			0
$841$	−9.74899	+	16.8857i	−0.336172	+	0.582267i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5.44597	+	9.43270i	0.187347	+	0.324495i
$846$		0			0
$847$	−6.98689	−	7.73990i	−0.240072	−	0.265946i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.91164	−	1.68104i	0.0998097	−	0.0576252i
$852$		0			0
$853$	20.8404	−	12.0322i	0.713560	−	0.411974i	−0.0988175	−	0.995106i	$-0.531506\pi$
$853$	20.8404	−	12.0322i	0.713560	−	0.411974i	0.812378	+	0.583131i	$0.198173\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	50.7738			1.73440			0.867200	−	0.497960i	$-0.165917\pi$
$857$	50.7738			1.73440			0.867200	+	0.497960i	$0.165917\pi$
$858$		0			0
$859$			7.83143i			0.267205i	0.991035	+	0.133603i	$0.0426545\pi$
$859$			7.83143i			0.267205i	−0.991035	+	0.133603i	$0.957345\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	3.68312	−	2.12645i	0.125375	−	0.0723852i	−0.436001	−	0.899946i	$-0.643606\pi$
$863$	3.68312	−	2.12645i	0.125375	−	0.0723852i	0.561376	+	0.827561i	$0.310272\pi$
$864$		0			0
$865$	12.1264	−	21.0036i	0.412311	−	0.714143i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−11.9019	−	6.87156i	−0.403744	−	0.233102i
$870$		0			0
$871$			4.44218i			0.150518i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−17.5114	−	19.3987i	−0.591994	−	0.655795i
$876$		0			0
$877$	34.1070			1.15171			0.575855	−	0.817552i	$-0.304669\pi$
$877$	34.1070			1.15171			0.575855	+	0.817552i	$0.304669\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−8.87180			−0.298898			−0.149449	−	0.988769i	$-0.547750\pi$
$881$	−8.87180			−0.298898			−0.149449	+	0.988769i	$0.547750\pi$
$882$		0			0
$883$	43.8791			1.47665			0.738325	−	0.674446i	$-0.235617\pi$
$883$	43.8791			1.47665			0.738325	+	0.674446i	$0.235617\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−36.7398			−1.23360			−0.616801	−	0.787119i	$-0.711572\pi$
$887$	−36.7398			−1.23360			−0.616801	+	0.787119i	$0.711572\pi$
$888$		0			0
$889$	−31.2307	−	34.5966i	−1.04744	−	1.16033i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			30.5448i			1.02214i
$894$		0			0
$895$	4.95206	+	2.85907i	0.165529	+	0.0955683i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.21502	−	2.10448i	0.0405232	−	0.0701883i
$900$		0			0
$901$	6.36603	−	3.67543i	0.212083	−	0.122446i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	13.2552i		−	0.440617i
$906$		0			0
$907$	−16.6579			−0.553118			−0.276559	−	0.960997i	$-0.589194\pi$
$907$	−16.6579			−0.553118			−0.276559	+	0.960997i	$0.589194\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−29.0963	+	16.7988i	−0.964004	+	0.556568i	−0.897403	−	0.441212i	$-0.854549\pi$
$911$	−29.0963	+	16.7988i	−0.964004	+	0.556568i	−0.0666011	+	0.997780i	$0.521215\pi$
$912$		0			0
$913$	−34.8317	+	20.1101i	−1.15276	+	0.665548i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	18.0805	+	20.0291i	0.597071	+	0.661420i
$918$		0			0
$919$	−5.96310	−	10.3284i	−0.196705	−	0.340702i	0.750753	−	0.660583i	$-0.229691\pi$
$919$	−5.96310	−	10.3284i	−0.196705	−	0.340702i	−0.947458	+	0.319880i	$0.896357\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−11.4041	+	19.7525i	−0.375371	+	0.650161i
$924$		0			0
$925$	6.43970	+	11.1539i	0.211736	+	0.366738i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−21.8590	+	37.8609i	−0.717170	+	1.24217i	0.244947	+	0.969536i	$0.421229\pi$
$929$	−21.8590	+	37.8609i	−0.717170	+	1.24217i	−0.962117	+	0.272638i	$0.912104\pi$
$930$		0			0
$931$	8.50331	+	18.9687i	0.278685	+	0.621673i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.39497	−	0.805388i	−0.0456205	−	0.0263390i
$936$		0			0
$937$			47.1189i			1.53931i	0.638461	+	0.769654i	$0.279571\pi$
$937$			47.1189i			1.53931i	−0.638461	+	0.769654i	$0.720429\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−11.0516	−	19.1420i	−0.360273	−	0.624011i	0.627733	−	0.778429i	$-0.283983\pi$
$941$	−11.0516	−	19.1420i	−0.360273	−	0.624011i	−0.988006	+	0.154418i	$0.950650\pi$
$942$		0			0
$943$	3.74333	+	2.16121i	0.121900	+	0.0703788i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.0288642	+	0.0166647i	0.000937959	+	0.000541531i	0.500469	−	0.865755i	$-0.333161\pi$
$947$	0.0288642	+	0.0166647i	0.000937959	+	0.000541531i	−0.499531	+	0.866296i	$0.666494\pi$
$948$		0			0
$949$	7.49946	+	12.9894i	0.243443	+	0.421655i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	21.6371i		−	0.700893i	−0.936583	−	0.350446i	$-0.886030\pi$
$953$		−	21.6371i		−	0.700893i	0.936583	−	0.350446i	$-0.113970\pi$
$954$		0			0
$955$	22.2224	+	12.8301i	0.719100	+	0.415173i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	16.5186	−	14.9115i	0.533414	−	0.481518i
$960$		0			0
$961$	−15.1893	+	26.3086i	−0.489976	+	0.848664i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.633837	−	1.09784i	−0.0204039	−	0.0353406i
$966$		0			0
$967$	−26.4627	+	45.8347i	−0.850983	+	1.47395i	0.0293390	+	0.999570i	$0.490660\pi$
$967$	−26.4627	+	45.8347i	−0.850983	+	1.47395i	−0.880322	+	0.474376i	$0.842674\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	8.18246	+	14.1724i	0.262588	+	0.454815i	0.966929	−	0.255047i	$-0.0820908\pi$
$971$	8.18246	+	14.1724i	0.262588	+	0.454815i	−0.704341	+	0.709862i	$0.748757\pi$
$972$		0			0
$973$	−5.64696	−	26.4016i	−0.181033	−	0.846397i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−25.2685	+	14.5888i	−0.808412	+	0.466737i	−0.846404	−	0.532541i	$-0.821237\pi$
$977$	−25.2685	+	14.5888i	−0.808412	+	0.466737i	0.0379920	+	0.999278i	$0.487904\pi$
$978$		0			0
$979$	−30.0476	+	17.3480i	−0.960324	+	0.554444i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.8109			0.536184			0.268092	−	0.963393i	$-0.413607\pi$
$983$	16.8109			0.536184			0.268092	+	0.963393i	$0.413607\pi$
$984$		0			0
$985$			3.61685i			0.115242i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.42707	−	0.823919i	0.0453782	−	0.0261991i
$990$		0			0
$991$	24.5748	−	42.5649i	0.780645	−	1.35212i	−0.150921	−	0.988546i	$-0.548224\pi$
$991$	24.5748	−	42.5649i	0.780645	−	1.35212i	0.931566	−	0.363571i	$-0.118443\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.392590	−	0.226662i	−0.0124459	−	0.00718567i
$996$		0			0
$997$		−	17.8806i		−	0.566284i	−0.959078	−	0.283142i	$-0.908623\pi$
$997$		−	17.8806i		−	0.566284i	0.959078	−	0.283142i	$-0.0913768\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.bm.j.1025.6		32
3.2	odd	2	inner	2268.2.bm.j.1025.11		32
7.5	odd	6		2268.2.w.j.1349.6		32
9.2	odd	6		2268.2.w.j.269.6		32
9.4	even	3		2268.2.t.c.1781.11	yes	32
9.5	odd	6		2268.2.t.c.1781.6	✓	32
9.7	even	3		2268.2.w.j.269.11		32
21.5	even	6		2268.2.w.j.1349.11		32
63.5	even	6		2268.2.t.c.2105.11	yes	32
63.40	odd	6		2268.2.t.c.2105.6	yes	32
63.47	even	6	inner	2268.2.bm.j.593.6		32
63.61	odd	6	inner	2268.2.bm.j.593.11		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.t.c.1781.6	✓	32	9.5	odd	6
2268.2.t.c.1781.11	yes	32	9.4	even	3
2268.2.t.c.2105.6	yes	32	63.40	odd	6
2268.2.t.c.2105.11	yes	32	63.5	even	6
2268.2.w.j.269.6		32	9.2	odd	6
2268.2.w.j.269.11		32	9.7	even	3
2268.2.w.j.1349.6		32	7.5	odd	6
2268.2.w.j.1349.11		32	21.5	even	6
2268.2.bm.j.593.6		32	63.47	even	6	inner
2268.2.bm.j.593.11		32	63.61	odd	6	inner
2268.2.bm.j.1025.6		32	1.1	even	1	trivial
2268.2.bm.j.1025.11		32	3.2	odd	2	inner

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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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.bm (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-1.13332 q^{5} +(-1.77285 - 1.96392i) q^{7} +2.65687i q^{11} +(1.59436 + 0.920505i) q^{13} +(0.267475 - 0.463281i) q^{17} +(2.57178 - 1.48482i) q^{19} +0.969928i q^{23} -3.71559 q^{25} +(-2.66955 + 1.54127i) q^{29} +(-0.682710 + 0.394163i) q^{31} +(2.00920 + 2.22575i) q^{35} +(-1.73316 - 3.00191i) q^{37} +(2.22822 - 3.85939i) q^{41} +(-0.849465 - 1.47132i) q^{43} +(-5.14286 + 8.90769i) q^{47} +(-0.713980 + 6.96349i) q^{49} +(11.9002 + 6.87059i) q^{53} -3.01107i q^{55} +(3.64838 + 6.31917i) q^{59} +(-1.74194 - 1.00571i) q^{61} +(-1.80692 - 1.04322i) q^{65} +(1.20645 + 2.08964i) q^{67} +12.3890i q^{71} +(7.05561 + 4.07356i) q^{73} +(5.21789 - 4.71024i) q^{77} +(-2.58634 + 4.47967i) q^{79} +(7.56910 + 13.1101i) q^{83} +(-0.303134 + 0.525044i) q^{85} +(6.52948 + 11.3094i) q^{89} +(-1.01877 - 4.76312i) q^{91} +(-2.91464 + 1.68277i) q^{95} +(1.44234 - 0.832736i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 4 q^{7} - 12 q^{13} + 32 q^{25} - 24 q^{31} - 4 q^{37} - 4 q^{43} - 16 q^{49} - 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} + 12 q^{85} - 36 q^{91} - 12 q^{97}+O(q^{100}) \) 32 * q + 4 * q^7 - 12 * q^13 + 32 * q^25 - 24 * q^31 - 4 * q^37 - 4 * q^43 - 16 * q^49 - 12 * q^61 + 4 * q^67 - 36 * q^73 + 28 * q^79 + 12 * q^85 - 36 * q^91 - 12 * q^97

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