LMFDB - Embedded newform 2736.2.bm.i.559.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(559,2736)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2736.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.bm (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 912)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			559.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2736.559
Dual form			2736.2.bm.i.1855.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{5} -4.56048i q^{7} +O(q^{10})$	$q+(-1.00000 + 1.73205i) q^{5} -4.56048i q^{7} -5.65685i q^{11} +(3.94949 - 2.28024i) q^{13} +(-1.00000 + 1.73205i) q^{17} +(-4.00000 - 1.73205i) q^{19} +(-4.89898 + 2.82843i) q^{23} +(0.500000 + 0.866025i) q^{25} +(1.89898 - 1.09638i) q^{29} -7.89898 q^{31} +(7.89898 + 4.56048i) q^{35} +1.09638i q^{37} +(-4.50000 - 2.59808i) q^{43} +(7.89898 - 4.56048i) q^{47} -13.7980 q^{49} +(-10.8990 + 6.29253i) q^{53} +(9.79796 + 5.65685i) q^{55} +(-5.89898 + 10.2173i) q^{59} +(4.94949 + 8.57277i) q^{61} +9.12096i q^{65} +(4.39898 + 7.61926i) q^{67} +(2.00000 - 3.46410i) q^{71} +(3.39898 - 5.88721i) q^{73} -25.7980 q^{77} +(0.0505103 - 0.0874863i) q^{79} +3.46410i q^{83} +(-2.00000 - 3.46410i) q^{85} +(1.89898 - 1.09638i) q^{89} +(-10.3990 - 18.0116i) q^{91} +(7.00000 - 5.19615i) q^{95} +(3.79796 + 2.19275i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5}+O(q^{10}) $ 4 * q - 4 * q^5	$ 4 q - 4 q^{5} + 6 q^{13} - 4 q^{17} - 16 q^{19} + 2 q^{25} - 12 q^{29} - 12 q^{31} + 12 q^{35} - 18 q^{43} + 12 q^{47} - 16 q^{49} - 24 q^{53} - 4 q^{59} + 10 q^{61} - 2 q^{67} + 8 q^{71} - 6 q^{73} - 64 q^{77} + 10 q^{79} - 8 q^{85} - 12 q^{89} - 22 q^{91} + 28 q^{95} - 24 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 + 6 * q^13 - 4 * q^17 - 16 * q^19 + 2 * q^25 - 12 * q^29 - 12 * q^31 + 12 * q^35 - 18 * q^43 + 12 * q^47 - 16 * q^49 - 24 * q^53 - 4 * q^59 + 10 * q^61 - 2 * q^67 + 8 * q^71 - 6 * q^73 - 64 * q^77 + 10 * q^79 - 8 * q^85 - 12 * q^89 - 22 * q^91 + 28 * q^95 - 24 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$-1$	$1$

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
$5$	−1.00000	+	1.73205i	−0.447214	+	0.774597i	−0.998203	−	0.0599153i	$-0.980917\pi$
$5$	−1.00000	+	1.73205i	−0.447214	+	0.774597i	0.550990	+	0.834512i	$0.314250\pi$
$6$		0			0
$7$		−	4.56048i		−	1.72370i	−0.507164	−	0.861849i	$-0.669306\pi$
$7$		−	4.56048i		−	1.72370i	0.507164	−	0.861849i	$-0.330694\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.65685i		−	1.70561i	−0.522233	−	0.852803i	$-0.674901\pi$
$11$		−	5.65685i		−	1.70561i	0.522233	−	0.852803i	$-0.325099\pi$
$12$		0			0
$13$	3.94949	−	2.28024i	1.09539	−	0.632425i	0.160385	−	0.987055i	$-0.448727\pi$
$13$	3.94949	−	2.28024i	1.09539	−	0.632425i	0.935007	+	0.354630i	$0.115393\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	−0.961436	−	0.275029i	$-0.911312\pi$
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	0.718900	+	0.695113i	$0.244646\pi$
$18$		0			0
$19$	−4.00000	−	1.73205i	−0.917663	−	0.397360i
$19$	−4.00000	−	1.73205i	−0.917663	−	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.89898	+	2.82843i	−1.02151	+	0.589768i	−0.914540	−	0.404495i	$-0.867447\pi$
$23$	−4.89898	+	2.82843i	−1.02151	+	0.589768i	−0.106967	+	0.994263i	$0.534114\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.89898	−	1.09638i	0.352632	−	0.203592i	−0.313212	−	0.949683i	$-0.601405\pi$
$29$	1.89898	−	1.09638i	0.352632	−	0.203592i	0.665844	+	0.746091i	$0.268072\pi$
$30$		0			0
$31$	−7.89898			−1.41870			−0.709349	−	0.704857i	$-0.751011\pi$
$31$	−7.89898			−1.41870			−0.709349	+	0.704857i	$0.751011\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	7.89898	+	4.56048i	1.33517	+	0.770861i
$36$		0			0
$37$			1.09638i			0.180243i	0.995931	+	0.0901216i	$0.0287256\pi$
$37$			1.09638i			0.180243i	−0.995931	+	0.0901216i	$0.971274\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	−4.50000	−	2.59808i	−0.686244	−	0.396203i	0.115960	−	0.993254i	$-0.463006\pi$
$43$	−4.50000	−	2.59808i	−0.686244	−	0.396203i	−0.802203	+	0.597051i	$0.796339\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.89898	−	4.56048i	1.15218	−	0.665214i	0.202766	−	0.979227i	$-0.435007\pi$
$47$	7.89898	−	4.56048i	1.15218	−	0.665214i	0.949419	+	0.314013i	$0.101674\pi$
$48$		0			0
$49$	−13.7980			−1.97114
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.8990	+	6.29253i	−1.49709	+	0.864345i	−0.999994	−	0.00335049i	$-0.998934\pi$
$53$	−10.8990	+	6.29253i	−1.49709	+	0.864345i	−0.497096	+	0.867696i	$0.665600\pi$
$54$		0			0
$55$	9.79796	+	5.65685i	1.32116	+	0.762770i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.89898	+	10.2173i	−0.767982	+	1.33018i	0.170674	+	0.985328i	$0.445406\pi$
$59$	−5.89898	+	10.2173i	−0.767982	+	1.33018i	−0.938656	+	0.344856i	$0.887928\pi$
$60$		0			0
$61$	4.94949	+	8.57277i	0.633717	+	1.09763i	0.986785	+	0.162033i	$0.0518050\pi$
$61$	4.94949	+	8.57277i	0.633717	+	1.09763i	−0.353068	+	0.935598i	$0.614862\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			9.12096i			1.13132i
$66$		0			0
$67$	4.39898	+	7.61926i	0.537421	+	0.930840i	0.999042	+	0.0437630i	$0.0139347\pi$
$67$	4.39898	+	7.61926i	0.537421	+	0.930840i	−0.461621	+	0.887077i	$0.652732\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.00000	−	3.46410i	0.237356	−	0.411113i	−0.722599	−	0.691268i	$-0.757052\pi$
$71$	2.00000	−	3.46410i	0.237356	−	0.411113i	0.959955	+	0.280155i	$0.0903858\pi$
$72$		0			0
$73$	3.39898	−	5.88721i	0.397820	−	0.689045i	−0.595636	−	0.803254i	$-0.703100\pi$
$73$	3.39898	−	5.88721i	0.397820	−	0.689045i	0.993457	+	0.114209i	$0.0364334\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−25.7980			−2.93995
$78$		0			0
$79$	0.0505103	−	0.0874863i	0.00568285	−	0.00984298i	−0.863170	−	0.504913i	$-0.831524\pi$
$79$	0.0505103	−	0.0874863i	0.00568285	−	0.00984298i	0.868853	+	0.495070i	$0.164858\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.46410i			0.380235i	0.981761	+	0.190117i	$0.0608868\pi$
$83$			3.46410i			0.380235i	−0.981761	+	0.190117i	$0.939113\pi$
$84$		0			0
$85$	−2.00000	−	3.46410i	−0.216930	−	0.375735i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.89898	−	1.09638i	0.201291	−	0.116216i	−0.395966	−	0.918265i	$-0.629590\pi$
$89$	1.89898	−	1.09638i	0.201291	−	0.116216i	0.597258	+	0.802049i	$0.296257\pi$
$90$		0			0
$91$	−10.3990	−	18.0116i	−1.09011	−	1.88812i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.00000	−	5.19615i	0.718185	−	0.533114i
$96$		0			0
$97$	3.79796	+	2.19275i	0.385624	+	0.222640i	0.680262	−	0.732969i	$-0.261866\pi$
$97$	3.79796	+	2.19275i	0.385624	+	0.222640i	−0.294638	+	0.955609i	$0.595199\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.00000	−	10.3923i	−0.597022	−	1.03407i	−0.993258	−	0.115924i	$-0.963017\pi$
$101$	−6.00000	−	10.3923i	−0.597022	−	1.03407i	0.396236	−	0.918149i	$-0.370316\pi$
$102$		0			0
$103$	2.10102			0.207020			0.103510	−	0.994628i	$-0.466993\pi$
$103$	2.10102			0.207020			0.103510	+	0.994628i	$0.466993\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.00000			−0.773389			−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000			−0.773389			−0.386695	+	0.922208i	$0.626383\pi$
$108$		0			0
$109$	−14.6969	−	8.48528i	−1.40771	−	0.812743i	−0.412544	−	0.910938i	$-0.635360\pi$
$109$	−14.6969	−	8.48528i	−1.40771	−	0.812743i	−0.995167	+	0.0981950i	$0.968693\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			4.73545i			0.445474i	0.974879	+	0.222737i	$0.0714991\pi$
$113$			4.73545i			0.445474i	−0.974879	+	0.222737i	$0.928501\pi$
$114$		0			0
$115$		−	11.3137i		−	1.05501i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.89898	+	4.56048i	0.724098	+	0.418058i
$120$		0			0
$121$	−21.0000			−1.90909
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.0000			−1.07331
$126$		0			0
$127$	−2.89898	−	5.02118i	−0.257243	−	0.445558i	0.708259	−	0.705952i	$-0.249481\pi$
$127$	−2.89898	−	5.02118i	−0.257243	−	0.445558i	−0.965502	+	0.260395i	$0.916147\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−12.0000	−	6.92820i	−1.04844	−	0.605320i	−0.126231	−	0.992001i	$-0.540288\pi$
$131$	−12.0000	−	6.92820i	−1.04844	−	0.605320i	−0.922214	+	0.386681i	$0.873621\pi$
$132$		0			0
$133$	−7.89898	+	18.2419i	−0.684928	+	1.58177i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	−0.999893	−	0.0146279i	$-0.995344\pi$
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	0.487278	−	0.873247i	$-0.337990\pi$
$138$		0			0
$139$	−3.39898	+	1.96240i	−0.288298	+	0.166449i	−0.637174	−	0.770720i	$-0.719897\pi$
$139$	−3.39898	+	1.96240i	−0.288298	+	0.166449i	0.348876	+	0.937169i	$0.386563\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−12.8990	−	22.3417i	−1.07867	−	1.86831i
$144$		0			0
$145$			4.38551i			0.364196i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.89898	−	11.9494i	0.565186	−	0.978932i	−0.431846	−	0.901947i	$-0.642138\pi$
$149$	6.89898	−	11.9494i	0.565186	−	0.978932i	0.997032	−	0.0769842i	$-0.0245291\pi$
$150$		0			0
$151$	21.7980			1.77389			0.886946	−	0.461872i	$-0.152822\pi$
$151$	21.7980			1.77389			0.886946	+	0.461872i	$0.152822\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.89898	−	13.6814i	0.634461	−	1.09892i
$156$		0			0
$157$	7.84847	−	13.5939i	0.626376	−	1.08492i	−0.361897	−	0.932218i	$-0.617871\pi$
$157$	7.84847	−	13.5939i	0.626376	−	1.08492i	0.988273	−	0.152697i	$-0.0487959\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12.8990	+	22.3417i	1.01658	+	1.76077i
$162$		0			0
$163$			15.5885i			1.22098i	0.792023	+	0.610491i	$0.209028\pi$
$163$			15.5885i			1.22098i	−0.792023	+	0.610491i	$0.790972\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.00000	−	5.19615i	−0.232147	−	0.402090i	0.726293	−	0.687386i	$-0.241242\pi$
$167$	−3.00000	−	5.19615i	−0.232147	−	0.402090i	−0.958440	+	0.285295i	$0.907908\pi$
$168$		0			0
$169$	3.89898	−	6.75323i	0.299921	−	0.519479i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.89898	+	1.09638i	0.144377	+	0.0833559i	0.570448	−	0.821334i	$-0.306769\pi$
$173$	1.89898	+	1.09638i	0.144377	+	0.0833559i	−0.426072	+	0.904689i	$0.640103\pi$
$174$		0			0
$175$	3.94949	−	2.28024i	0.298553	−	0.172370i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−25.7980			−1.92823			−0.964115	−	0.265485i	$-0.914468\pi$
$179$	−25.7980			−1.92823			−0.964115	+	0.265485i	$0.914468\pi$
$180$		0			0
$181$	14.6969	−	8.48528i	1.09241	−	0.630706i	0.158196	−	0.987408i	$-0.449432\pi$
$181$	14.6969	−	8.48528i	1.09241	−	0.630706i	0.934218	+	0.356702i	$0.116099\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.89898	−	1.09638i	−0.139616	−	0.0806072i
$186$		0			0
$187$	9.79796	+	5.65685i	0.716498	+	0.413670i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	24.2487i		−	1.75458i	−0.479965	−	0.877288i	$-0.659351\pi$
$191$		−	24.2487i		−	1.75458i	0.479965	−	0.877288i	$-0.340649\pi$
$192$		0			0
$193$	−10.5000	−	6.06218i	−0.755807	−	0.436365i	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	−10.5000	−	6.06218i	−0.755807	−	0.436365i	−0.827788	+	0.561041i	$0.810401\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$	11.0505	−	6.38002i	0.783350	−	0.452267i	−0.0542663	−	0.998527i	$-0.517282\pi$
$199$	11.0505	−	6.38002i	0.783350	−	0.452267i	0.837616	+	0.546259i	$0.183949\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.00000	−	8.66025i	−0.350931	−	0.607831i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−9.79796	+	22.6274i	−0.677739	+	1.56517i
$210$		0			0
$211$	−10.3990	+	18.0116i	−0.715895	+	1.23997i	0.246718	+	0.969087i	$0.420648\pi$
$211$	−10.3990	+	18.0116i	−0.715895	+	1.23997i	−0.962613	+	0.270880i	$0.912685\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	9.00000	−	5.19615i	0.613795	−	0.354375i
$216$		0			0
$217$			36.0231i			2.44541i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			9.12096i			0.613542i
$222$		0			0
$223$	2.94949	−	5.10867i	0.197512	−	0.342102i	−0.750209	−	0.661201i	$-0.770047\pi$
$223$	2.94949	−	5.10867i	0.197512	−	0.342102i	0.947721	+	0.319099i	$0.103380\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.7980			−1.04855			−0.524274	−	0.851550i	$-0.675663\pi$
$227$	−15.7980			−1.04855			−0.524274	+	0.851550i	$0.675663\pi$
$228$		0			0
$229$	−7.69694			−0.508628			−0.254314	−	0.967122i	$-0.581850\pi$
$229$	−7.69694			−0.508628			−0.254314	+	0.967122i	$0.581850\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.79796	−	16.9706i	0.641886	−	1.11178i	−0.343126	−	0.939289i	$-0.611486\pi$
$233$	9.79796	−	16.9706i	0.641886	−	1.11178i	0.985011	−	0.172489i	$-0.0551809\pi$
$234$		0			0
$235$			18.2419i			1.18997i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			21.7060i			1.40405i	0.712155	+	0.702023i	$0.247719\pi$
$239$			21.7060i			1.40405i	−0.712155	+	0.702023i	$0.752281\pi$
$240$		0			0
$241$	−14.2980	+	8.25493i	−0.921013	+	0.531747i	−0.883958	−	0.467566i	$-0.845131\pi$
$241$	−14.2980	+	8.25493i	−0.921013	+	0.531747i	−0.0370546	+	0.999313i	$0.511798\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	13.7980	−	23.8988i	0.881519	−	1.52684i
$246$		0			0
$247$	−19.7474	+	2.28024i	−1.25650	+	0.145088i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.797959	−	0.460702i	0.0503667	−	0.0290792i	−0.474605	−	0.880199i	$-0.657409\pi$
$251$	0.797959	−	0.460702i	0.0503667	−	0.0290792i	0.524972	+	0.851120i	$0.324076\pi$
$252$		0			0
$253$	16.0000	+	27.7128i	1.00591	+	1.74229i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.8990	−	6.29253i	0.679860	−	0.392517i	−0.119943	−	0.992781i	$-0.538271\pi$
$257$	10.8990	−	6.29253i	0.679860	−	0.392517i	0.799802	+	0.600264i	$0.204938\pi$
$258$		0			0
$259$	5.00000			0.310685
$260$		0			0
$261$		0			0
$262$		0			0
$263$	15.7980	+	9.12096i	0.974144	+	0.562422i	0.900497	−	0.434862i	$-0.143203\pi$
$263$	15.7980	+	9.12096i	0.974144	+	0.562422i	0.0736468	+	0.997284i	$0.476536\pi$
$264$		0			0
$265$		−	25.1701i		−	1.54619i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−18.7980	−	10.8530i	−1.14613	−	0.661719i	−0.198190	−	0.980164i	$-0.563506\pi$
$269$	−18.7980	−	10.8530i	−1.14613	−	0.661719i	−0.947942	+	0.318444i	$0.896840\pi$
$270$		0			0
$271$	6.79796	+	3.92480i	0.412947	+	0.238415i	0.692055	−	0.721845i	$-0.256706\pi$
$271$	6.79796	+	3.92480i	0.412947	+	0.238415i	−0.279108	+	0.960260i	$0.590039\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.89898	−	2.82843i	0.295420	−	0.170561i
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	25.8990	−	14.9528i	1.54500	−	0.892008i	0.546492	−	0.837464i	$-0.315963\pi$
$281$	25.8990	−	14.9528i	1.54500	−	0.892008i	0.998511	−	0.0545441i	$-0.0173705\pi$
$282$		0			0
$283$	−6.79796	−	3.92480i	−0.404097	−	0.233305i	0.284153	−	0.958779i	$-0.408287\pi$
$283$	−6.79796	−	3.92480i	−0.404097	−	0.233305i	−0.688250	+	0.725473i	$0.741621\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			27.3629i			1.59856i	0.600962	+	0.799278i	$0.294784\pi$
$293$			27.3629i			1.59856i	−0.600962	+	0.799278i	$0.705216\pi$
$294$		0			0
$295$	−11.7980	−	20.4347i	−0.686904	−	1.18975i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.8990	+	22.3417i	−0.745967	+	1.29205i
$300$		0			0
$301$	−11.8485	+	20.5222i	−0.682934	+	1.18288i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−19.7980			−1.13363
$306$		0			0
$307$	3.79796	−	6.57826i	0.216761	−	0.375441i	−0.737055	−	0.675833i	$-0.763784\pi$
$307$	3.79796	−	6.57826i	0.216761	−	0.375441i	0.953816	+	0.300392i	$0.0971174\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	12.2351i		−	0.693790i	−0.937904	−	0.346895i	$-0.887236\pi$
$311$		−	12.2351i		−	0.693790i	0.937904	−	0.346895i	$-0.112764\pi$
$312$		0			0
$313$	−10.7980	−	18.7026i	−0.610337	−	1.05713i	−0.991184	−	0.132496i	$-0.957701\pi$
$313$	−10.7980	−	18.7026i	−0.610337	−	1.05713i	0.380847	−	0.924638i	$-0.375633\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.79796	+	3.92480i	−0.381811	+	0.220439i	−0.678606	−	0.734502i	$-0.737416\pi$
$317$	−6.79796	+	3.92480i	−0.381811	+	0.220439i	0.296795	+	0.954941i	$0.404082\pi$
$318$		0			0
$319$	−6.20204	−	10.7423i	−0.347248	−	0.601451i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7.00000	−	5.19615i	0.389490	−	0.289122i
$324$		0			0
$325$	3.94949	+	2.28024i	0.219078	+	0.126485i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−20.7980	−	36.0231i	−1.14663	−	1.98602i
$330$		0			0
$331$	20.7980			1.14316			0.571580	−	0.820547i	$-0.306331\pi$
$331$	20.7980			1.14316			0.571580	+	0.820547i	$0.306331\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−17.5959			−0.961368
$336$		0			0
$337$	7.19694	+	4.15515i	0.392042	+	0.226346i	0.683045	−	0.730377i	$-0.260655\pi$
$337$	7.19694	+	4.15515i	0.392042	+	0.226346i	−0.291002	+	0.956722i	$0.593989\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			44.6834i			2.41974i
$342$		0			0
$343$			31.0019i			1.67395i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.89898	+	4.56048i	0.424039	+	0.244819i	0.696804	−	0.717262i	$-0.254605\pi$
$347$	7.89898	+	4.56048i	0.424039	+	0.244819i	−0.272765	+	0.962081i	$0.587938\pi$
$348$		0			0
$349$	7.69694			0.412008			0.206004	−	0.978551i	$-0.433954\pi$
$349$	7.69694			0.412008			0.206004	+	0.978551i	$0.433954\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−15.7980			−0.840841			−0.420420	−	0.907329i	$-0.638117\pi$
$353$	−15.7980			−0.840841			−0.420420	+	0.907329i	$0.638117\pi$
$354$		0			0
$355$	4.00000	+	6.92820i	0.212298	+	0.367711i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.79796	+	5.65685i	0.517116	+	0.298557i	0.735754	−	0.677249i	$-0.236828\pi$
$359$	9.79796	+	5.65685i	0.517116	+	0.298557i	−0.218638	+	0.975806i	$0.570161\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.79796	+	11.7744i	0.355821	+	0.616301i
$366$		0			0
$367$	8.84847	−	5.10867i	0.461886	−	0.266670i	−0.250951	−	0.968000i	$-0.580743\pi$
$367$	8.84847	−	5.10867i	0.461886	−	0.266670i	0.712837	+	0.701330i	$0.247410\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	28.6969	+	49.7046i	1.48987	+	2.58053i
$372$		0			0
$373$		−	22.0560i		−	1.14201i	−0.820945	−	0.571007i	$-0.806553\pi$
$373$		−	22.0560i		−	1.14201i	0.820945	−	0.571007i	$-0.193447\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5.00000	−	8.66025i	0.257513	−	0.446026i
$378$		0			0
$379$	−32.5959			−1.67434			−0.837170	−	0.546943i	$-0.815791\pi$
$379$	−32.5959			−1.67434			−0.837170	+	0.546943i	$0.815791\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.89898	+	5.02118i	−0.148131	+	0.256570i	−0.930537	−	0.366199i	$-0.880659\pi$
$383$	−2.89898	+	5.02118i	−0.148131	+	0.256570i	0.782406	+	0.622769i	$0.213992\pi$
$384$		0			0
$385$	25.7980	−	44.6834i	1.31479	−	2.27728i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.6969	−	18.5276i	−0.542356	−	0.939389i	−0.998768	−	0.0496201i	$-0.984199\pi$
$389$	−10.6969	−	18.5276i	−0.542356	−	0.939389i	0.456412	−	0.889769i	$-0.349134\pi$
$390$		0			0
$391$		−	11.3137i		−	0.572159i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.101021	+	0.174973i	0.00508289	+	0.00880383i
$396$		0			0
$397$	7.94949	−	13.7689i	0.398973	−	0.691042i	−0.594626	−	0.804002i	$-0.702700\pi$
$397$	7.94949	−	13.7689i	0.398973	−	0.691042i	0.993600	+	0.112960i	$0.0360332\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13.8990	+	8.02458i	0.694082	+	0.400728i	0.805139	−	0.593086i	$-0.202090\pi$
$401$	13.8990	+	8.02458i	0.694082	+	0.400728i	−0.111057	+	0.993814i	$0.535424\pi$
$402$		0			0
$403$	−31.1969	+	18.0116i	−1.55403	+	0.897220i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.20204			0.307424
$408$		0			0
$409$	−6.00000	+	3.46410i	−0.296681	+	0.171289i	−0.640951	−	0.767582i	$-0.721460\pi$
$409$	−6.00000	+	3.46410i	−0.296681	+	0.171289i	0.344270	+	0.938871i	$0.388126\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	46.5959	+	26.9022i	2.29284	+	1.32377i
$414$		0			0
$415$	−6.00000	−	3.46410i	−0.294528	−	0.170046i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	21.3561i		−	1.04331i	−0.853156	−	0.521656i	$-0.825314\pi$
$419$		−	21.3561i		−	1.04331i	0.853156	−	0.521656i	$-0.174686\pi$
$420$		0			0
$421$	−4.89898	−	2.82843i	−0.238762	−	0.137849i	0.375846	−	0.926682i	$-0.377352\pi$
$421$	−4.89898	−	2.82843i	−0.238762	−	0.137849i	−0.614607	+	0.788833i	$0.710686\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.00000			−0.0970143
$426$		0			0
$427$	39.0959	−	22.5720i	1.89198	−	1.09234i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.89898	+	3.28913i	0.0914706	+	0.158432i	0.908130	−	0.418688i	$-0.137510\pi$
$431$	1.89898	+	3.28913i	0.0914706	+	0.158432i	−0.816660	+	0.577120i	$0.804177\pi$
$432$		0			0
$433$	−19.1969	+	11.0834i	−0.922546	+	0.532632i	−0.884446	−	0.466642i	$-0.845464\pi$
$433$	−19.1969	+	11.0834i	−0.922546	+	0.532632i	−0.0380996	+	0.999274i	$0.512130\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	24.4949	−	2.82843i	1.17175	−	0.135302i
$438$		0			0
$439$	−3.84847	+	6.66574i	−0.183677	+	0.318139i	−0.943130	−	0.332424i	$-0.892134\pi$
$439$	−3.84847	+	6.66574i	−0.183677	+	0.318139i	0.759453	+	0.650563i	$0.225467\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.797959	+	0.460702i	−0.0379122	+	0.0218886i	−0.518836	−	0.854874i	$-0.673635\pi$
$443$	−0.797959	+	0.460702i	−0.0379122	+	0.0218886i	0.480924	+	0.876762i	$0.340301\pi$
$444$		0			0
$445$			4.38551i			0.207893i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			6.00680i			0.283478i	0.989904	+	0.141739i	$0.0452694\pi$
$449$			6.00680i			0.283478i	−0.989904	+	0.141739i	$0.954731\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	41.5959			1.95005
$456$		0			0
$457$	1.20204			0.0562291			0.0281146	−	0.999605i	$-0.491050\pi$
$457$	1.20204			0.0562291			0.0281146	+	0.999605i	$0.491050\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8.79796	−	15.2385i	0.409762	−	0.709728i	−0.585101	−	0.810960i	$-0.698945\pi$
$461$	8.79796	−	15.2385i	0.409762	−	0.709728i	0.994863	+	0.101232i	$0.0322785\pi$
$462$		0			0
$463$		−	13.6814i		−	0.635830i	−0.948119	−	0.317915i	$-0.897017\pi$
$463$		−	13.6814i		−	0.635830i	0.948119	−	0.317915i	$-0.102983\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			39.0265i			1.80593i	0.429712	+	0.902966i	$0.358615\pi$
$467$			39.0265i			1.80593i	−0.429712	+	0.902966i	$0.641385\pi$
$468$		0			0
$469$	34.7474	−	20.0614i	1.60449	−	0.926352i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−14.6969	+	25.4558i	−0.675766	+	1.17046i
$474$		0			0
$475$	−0.500000	−	4.33013i	−0.0229416	−	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.20204	+	1.27135i	−0.100614	+	0.0580894i	−0.549463	−	0.835518i	$-0.685168\pi$
$479$	−2.20204	+	1.27135i	−0.100614	+	0.0580894i	0.448849	+	0.893608i	$0.351834\pi$
$480$		0			0
$481$	2.50000	+	4.33013i	0.113990	+	0.197437i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.59592	+	4.38551i	−0.344913	+	0.199136i
$486$		0			0
$487$	−17.7980			−0.806503			−0.403251	−	0.915089i	$-0.632120\pi$
$487$	−17.7980			−0.806503			−0.403251	+	0.915089i	$0.632120\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.20204	+	1.27135i	0.0993767	+	0.0573752i	0.548865	−	0.835911i	$-0.315060\pi$
$491$	2.20204	+	1.27135i	0.0993767	+	0.0573752i	−0.449488	+	0.893286i	$0.648394\pi$
$492$		0			0
$493$			4.38551i			0.197513i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−15.7980	−	9.12096i	−0.708635	−	0.409131i
$498$		0			0
$499$	31.1969	+	18.0116i	1.39657	+	0.806308i	0.994031	−	0.109096i	$-0.0347956\pi$
$499$	31.1969	+	18.0116i	1.39657	+	0.806308i	0.402536	+	0.915404i	$0.368129\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5.69694	+	3.28913i	−0.254014	+	0.146655i	−0.621601	−	0.783334i	$-0.713517\pi$
$503$	−5.69694	+	3.28913i	−0.254014	+	0.146655i	0.367587	+	0.929989i	$0.380184\pi$
$504$		0			0
$505$	24.0000			1.06799
$506$		0			0
$507$		0			0
$508$		0			0
$509$	22.1010	−	12.7600i	0.979611	−	0.565578i	0.0774580	−	0.996996i	$-0.475320\pi$
$509$	22.1010	−	12.7600i	0.979611	−	0.565578i	0.902153	+	0.431417i	$0.141986\pi$
$510$		0			0
$511$	−26.8485	−	15.5010i	−1.18771	−	0.685723i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.10102	+	3.63907i	−0.0925820	+	0.160357i
$516$		0			0
$517$	−25.7980	−	44.6834i	−1.13459	−	1.96517i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	12.5851i		−	0.551361i	−0.961249	−	0.275681i	$-0.911097\pi$
$521$		−	12.5851i		−	0.551361i	0.961249	−	0.275681i	$-0.0889032\pi$
$522$		0			0
$523$	6.29796	+	10.9084i	0.275391	+	0.476990i	0.970234	−	0.242171i	$-0.0778595\pi$
$523$	6.29796	+	10.9084i	0.275391	+	0.476990i	−0.694843	+	0.719161i	$0.744526\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.89898	−	13.6814i	0.344085	−	0.595973i
$528$		0			0
$529$	4.50000	−	7.79423i	0.195652	−	0.338880i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	8.00000	−	13.8564i	0.345870	−	0.599065i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			78.0530i			3.36198i
$540$		0			0
$541$	15.9495	+	27.6253i	0.685722	+	1.18771i	0.973209	+	0.229921i	$0.0738468\pi$
$541$	15.9495	+	27.6253i	0.685722	+	1.18771i	−0.287487	+	0.957784i	$0.592820\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	29.3939	−	16.9706i	1.25910	−	0.726939i
$546$		0			0
$547$	−19.5000	−	33.7750i	−0.833760	−	1.44411i	−0.895036	−	0.445993i	$-0.852850\pi$
$547$	−19.5000	−	33.7750i	−0.833760	−	1.44411i	0.0612764	−	0.998121i	$-0.480483\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−9.49490	+	1.09638i	−0.404496	+	0.0467072i
$552$		0			0
$553$	−0.398979	−	0.230351i	−0.0169663	−	0.00979552i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12.0000	−	20.7846i	−0.508456	−	0.880672i	−0.999952	−	0.00979220i	$-0.996883\pi$
$557$	−12.0000	−	20.7846i	−0.508456	−	0.880672i	0.491496	−	0.870880i	$-0.336450\pi$
$558$		0			0
$559$	−23.6969			−1.00227
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.40408			−0.185610			−0.0928050	−	0.995684i	$-0.529583\pi$
$563$	−4.40408			−0.185610			−0.0928050	+	0.995684i	$0.529583\pi$
$564$		0			0
$565$	−8.20204	−	4.73545i	−0.345062	−	0.199222i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			25.1701i			1.05519i	0.849497	+	0.527593i	$0.176905\pi$
$569$			25.1701i			1.05519i	−0.849497	+	0.527593i	$0.823095\pi$
$570$		0			0
$571$		−	21.2453i		−	0.889089i	−0.895757	−	0.444544i	$-0.853366\pi$
$571$		−	21.2453i		−	0.889089i	0.895757	−	0.444544i	$-0.146634\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.89898	−	2.82843i	−0.204302	−	0.117954i
$576$		0			0
$577$	−41.5959			−1.73166			−0.865830	−	0.500338i	$-0.833209\pi$
$577$	−41.5959			−1.73166			−0.865830	+	0.500338i	$0.833209\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	15.7980			0.655410
$582$		0			0
$583$	35.5959	+	61.6539i	1.47423	+	2.55345i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.8990	−	8.02458i	−0.573672	−	0.331210i	0.184942	−	0.982749i	$-0.440790\pi$
$587$	−13.8990	−	8.02458i	−0.573672	−	0.331210i	−0.758615	+	0.651540i	$0.774123\pi$
$588$		0			0
$589$	31.5959	+	13.6814i	1.30189	+	0.563734i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−7.89898	−	13.6814i	−0.324372	−	0.561829i	0.657013	−	0.753879i	$-0.271820\pi$
$593$	−7.89898	−	13.6814i	−0.324372	−	0.561829i	−0.981385	+	0.192050i	$0.938486\pi$
$594$		0			0
$595$	−15.7980	+	9.12096i	−0.647653	+	0.373923i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	11.1010	+	19.2275i	0.453575	+	0.785616i	0.998605	−	0.0528011i	$-0.0168149\pi$
$599$	11.1010	+	19.2275i	0.453575	+	0.785616i	−0.545030	+	0.838417i	$0.683482\pi$
$600$		0			0
$601$		−	38.5658i		−	1.57313i	−0.617506	−	0.786566i	$-0.711857\pi$
$601$		−	38.5658i		−	1.57313i	0.617506	−	0.786566i	$-0.288143\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	21.0000	−	36.3731i	0.853771	−	1.47878i
$606$		0			0
$607$	21.8990			0.888852			0.444426	−	0.895816i	$-0.353408\pi$
$607$	21.8990			0.888852			0.444426	+	0.895816i	$0.353408\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	20.7980	−	36.0231i	0.841395	−	1.45734i
$612$		0			0
$613$	−2.79796	+	4.84621i	−0.113008	+	0.195736i	−0.916982	−	0.398929i	$-0.869382\pi$
$613$	−2.79796	+	4.84621i	−0.113008	+	0.195736i	0.803973	+	0.594665i	$0.202715\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.89898	+	5.02118i	0.116709	+	0.202145i	0.918461	−	0.395511i	$-0.129432\pi$
$617$	2.89898	+	5.02118i	0.116709	+	0.202145i	−0.801753	+	0.597656i	$0.796099\pi$
$618$		0			0
$619$		−	1.38211i		−	0.0555515i	−0.999614	−	0.0277758i	$-0.991158\pi$
$619$		−	1.38211i		−	0.0555515i	0.999614	−	0.0277758i	$-0.00884243\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.00000	−	8.66025i	−0.200321	−	0.346966i
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.89898	−	1.09638i	−0.0757173	−	0.0437154i
$630$		0			0
$631$	−2.35357	+	1.35884i	−0.0936942	+	0.0540944i	−0.546115	−	0.837710i	$-0.683894\pi$
$631$	−2.35357	+	1.35884i	−0.0936942	+	0.0540944i	0.452421	+	0.891805i	$0.350561\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	11.5959			0.460170
$636$		0			0
$637$	−54.4949	+	31.4626i	−2.15917	+	1.24660i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	30.7980	+	17.7812i	1.21645	+	0.702316i	0.964156	−	0.265336i	$-0.0854829\pi$
$641$	30.7980	+	17.7812i	1.21645	+	0.702316i	0.252290	+	0.967652i	$0.418816\pi$
$642$		0			0
$643$	−24.0959	−	13.9118i	−0.950250	−	0.548627i	−0.0570916	−	0.998369i	$-0.518183\pi$
$643$	−24.0959	−	13.9118i	−0.950250	−	0.548627i	−0.893159	+	0.449742i	$0.851516\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	50.6902i		−	1.99284i	−0.0845621	−	0.996418i	$-0.526949\pi$
$647$		−	50.6902i		−	1.99284i	0.0845621	−	0.996418i	$-0.473051\pi$
$648$		0			0
$649$	57.7980	+	33.3697i	2.26877	+	1.30987i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.40408			0.172345			0.0861725	−	0.996280i	$-0.472536\pi$
$653$	4.40408			0.172345			0.0861725	+	0.996280i	$0.472536\pi$
$654$		0			0
$655$	24.0000	−	13.8564i	0.937758	−	0.541415i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10.6969	−	18.5276i	−0.416694	−	0.721734i	0.578911	−	0.815391i	$-0.303478\pi$
$659$	−10.6969	−	18.5276i	−0.416694	−	0.721734i	−0.995605	+	0.0936563i	$0.970145\pi$
$660$		0			0
$661$	6.49490	−	3.74983i	0.252622	−	0.145852i	−0.368342	−	0.929690i	$-0.620074\pi$
$661$	6.49490	−	3.74983i	0.252622	−	0.145852i	0.620964	+	0.783839i	$0.286741\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−23.6969	−	31.9233i	−0.918928	−	1.23793i
$666$		0			0
$667$	−6.20204	+	10.7423i	−0.240144	+	0.415942i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	48.4949	−	27.9985i	1.87212	−	1.08087i
$672$		0			0
$673$		−	37.8659i		−	1.45962i	−0.683648	−	0.729812i	$-0.739608\pi$
$673$		−	37.8659i		−	1.45962i	0.683648	−	0.729812i	$-0.260392\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			25.5201i			0.980816i	0.871493	+	0.490408i	$0.163152\pi$
$677$			25.5201i			0.980816i	−0.871493	+	0.490408i	$0.836848\pi$
$678$		0			0
$679$	10.0000	−	17.3205i	0.383765	−	0.664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	16.2020			0.619954			0.309977	−	0.950744i	$-0.399679\pi$
$683$	16.2020			0.619954			0.309977	+	0.950744i	$0.399679\pi$
$684$		0			0
$685$	24.0000			0.916993
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−28.6969	+	49.7046i	−1.09327	+	1.89359i
$690$		0			0
$691$			24.2487i			0.922464i	0.887279	+	0.461232i	$0.152592\pi$
$691$			24.2487i			0.922464i	−0.887279	+	0.461232i	$0.847408\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	7.84961i		−	0.297753i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	21.0000	−	36.3731i	0.793159	−	1.37379i	−0.130843	−	0.991403i	$-0.541768\pi$
$701$	21.0000	−	36.3731i	0.793159	−	1.37379i	0.924002	−	0.382389i	$-0.124898\pi$
$702$		0			0
$703$	1.89898	−	4.38551i	0.0716214	−	0.165402i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−47.3939	+	27.3629i	−1.78243	+	1.02909i
$708$		0			0
$709$	4.15153	+	7.19066i	0.155914	+	0.270051i	0.933391	−	0.358860i	$-0.116834\pi$
$709$	4.15153	+	7.19066i	0.155914	+	0.270051i	−0.777477	+	0.628911i	$0.783501\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	38.6969	−	22.3417i	1.44921	−	0.836703i
$714$		0			0
$715$	51.5959			1.92958
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−22.8990	−	13.2207i	−0.853988	−	0.493050i	0.00800654	−	0.999968i	$-0.497451\pi$
$719$	−22.8990	−	13.2207i	−0.853988	−	0.493050i	−0.861994	+	0.506918i	$0.830785\pi$
$720$		0			0
$721$		−	9.58166i		−	0.356840i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.89898	+	1.09638i	0.0705263	+	0.0407184i
$726$		0			0
$727$	−21.3434	−	12.3226i	−0.791582	−	0.457020i	0.0489374	−	0.998802i	$-0.484417\pi$
$727$	−21.3434	−	12.3226i	−0.791582	−	0.457020i	−0.840519	+	0.541782i	$0.817750\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	9.00000	−	5.19615i	0.332877	−	0.192187i
$732$		0			0
$733$	−2.40408			−0.0887968			−0.0443984	−	0.999014i	$-0.514137\pi$
$733$	−2.40408			−0.0887968			−0.0443984	+	0.999014i	$0.514137\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	43.1010	−	24.8844i	1.58765	−	0.916628i
$738$		0			0
$739$	−18.3990	−	10.6227i	−0.676817	−	0.390761i	0.121837	−	0.992550i	$-0.461121\pi$
$739$	−18.3990	−	10.6227i	−0.676817	−	0.390761i	−0.798655	+	0.601789i	$0.794455\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.8990	+	18.8776i	−0.399845	+	0.692551i	−0.993706	−	0.112016i	$-0.964269\pi$
$743$	−10.8990	+	18.8776i	−0.399845	+	0.692551i	0.593862	+	0.804567i	$0.297603\pi$
$744$		0			0
$745$	13.7980	+	23.8988i	0.505518	+	0.875583i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			36.4838i			1.33309i
$750$		0			0
$751$	−3.74745	−	6.49077i	−0.136746	−	0.236852i	0.789517	−	0.613729i	$-0.210331\pi$
$751$	−3.74745	−	6.49077i	−0.136746	−	0.236852i	−0.926263	+	0.376877i	$0.876998\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21.7980	+	37.7552i	−0.793309	+	1.37405i
$756$		0			0
$757$	8.74745	−	15.1510i	0.317931	−	0.550673i	−0.662125	−	0.749394i	$-0.730345\pi$
$757$	8.74745	−	15.1510i	0.317931	−	0.550673i	0.980056	+	0.198720i	$0.0636785\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.79796			−0.137676			−0.0688380	−	0.997628i	$-0.521929\pi$
$761$	−3.79796			−0.137676			−0.0688380	+	0.997628i	$0.521929\pi$
$762$		0			0
$763$	−38.6969	+	67.0251i	−1.40092	+	2.42647i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			53.8043i			1.94276i
$768$		0			0
$769$	23.0959	+	40.0033i	0.832860	+	1.44256i	0.895761	+	0.444536i	$0.146632\pi$
$769$	23.0959	+	40.0033i	0.832860	+	1.44256i	−0.0629008	+	0.998020i	$0.520035\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	35.6969	−	20.6096i	1.28393	−	0.741277i	0.306365	−	0.951914i	$-0.400887\pi$
$773$	35.6969	−	20.6096i	1.28393	−	0.741277i	0.977564	+	0.210637i	$0.0675538\pi$
$774$		0			0
$775$	−3.94949	−	6.84072i	−0.141870	−	0.245726i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−19.5959	−	11.3137i	−0.701197	−	0.404836i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	15.6969	+	27.1879i	0.560248	+	0.970378i
$786$		0			0
$787$	−4.59592			−0.163827			−0.0819134	−	0.996639i	$-0.526103\pi$
$787$	−4.59592			−0.163827			−0.0819134	+	0.996639i	$0.526103\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	21.5959			0.767862
$792$		0			0
$793$	39.0959	+	22.5720i	1.38834	+	0.801557i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	38.3266i		−	1.35760i	−0.734324	−	0.678799i	$-0.762501\pi$
$797$		−	38.3266i		−	1.35760i	0.734324	−	0.678799i	$-0.237499\pi$
$798$		0			0
$799$			18.2419i			0.645352i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−33.3031	−	19.2275i	−1.17524	−	0.678525i
$804$		0			0
$805$	−51.5959			−1.81852
$806$		0			0
$807$		0			0
$808$		0			0
$809$	15.3939			0.541220			0.270610	−	0.962689i	$-0.412775\pi$
$809$	15.3939			0.541220			0.270610	+	0.962689i	$0.412775\pi$
$810$		0			0
$811$	4.00000	+	6.92820i	0.140459	+	0.243282i	0.927670	−	0.373402i	$-0.121809\pi$
$811$	4.00000	+	6.92820i	0.140459	+	0.243282i	−0.787211	+	0.616684i	$0.788476\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−27.0000	−	15.5885i	−0.945769	−	0.546040i
$816$		0			0
$817$	13.5000	+	18.1865i	0.472305	+	0.636266i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.69694	−	9.86739i	−0.198825	−	0.344374i	0.749323	−	0.662205i	$-0.230379\pi$
$821$	−5.69694	−	9.86739i	−0.198825	−	0.344374i	−0.948148	+	0.317830i	$0.897046\pi$
$822$		0			0
$823$	20.3939	−	11.7744i	0.710886	−	0.410430i	−0.100503	−	0.994937i	$-0.532045\pi$
$823$	20.3939	−	11.7744i	0.710886	−	0.410430i	0.811389	+	0.584507i	$0.198712\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	3.20204	+	5.54610i	0.111346	+	0.192857i	0.916313	−	0.400462i	$-0.131151\pi$
$827$	3.20204	+	5.54610i	0.111346	+	0.192857i	−0.804967	+	0.593319i	$0.797817\pi$
$828$		0			0
$829$			2.71767i			0.0943886i	0.998886	+	0.0471943i	$0.0150280\pi$
$829$			2.71767i			0.0943886i	−0.998886	+	0.0471943i	$0.984972\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	13.7980	−	23.8988i	0.478071	−	0.828043i
$834$		0			0
$835$	12.0000			0.415277
$836$		0			0
$837$		0			0
$838$		0			0
$839$	3.00000	−	5.19615i	0.103572	−	0.179391i	−0.809582	−	0.587007i	$-0.800306\pi$
$839$	3.00000	−	5.19615i	0.103572	−	0.179391i	0.913154	+	0.407615i	$0.133640\pi$
$840$		0			0
$841$	−12.0959	+	20.9507i	−0.417101	+	0.722439i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.79796	+	13.5065i	0.268258	+	0.464636i
$846$		0			0
$847$			95.7700i			3.29070i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.10102	−	5.37113i	−0.106302	−	0.184120i
$852$		0			0
$853$	−11.6464	+	20.1722i	−0.398766	+	0.690683i	−0.993574	−	0.113185i	$-0.963895\pi$
$853$	−11.6464	+	20.1722i	−0.398766	+	0.690683i	0.594808	+	0.803868i	$0.297228\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7.40408	−	4.27475i	−0.252919	−	0.146023i	0.368181	−	0.929754i	$-0.379981\pi$
$857$	−7.40408	−	4.27475i	−0.252919	−	0.146023i	−0.621100	+	0.783731i	$0.713314\pi$
$858$		0			0
$859$	23.2980	−	13.4511i	0.794916	−	0.458945i	−0.0467743	−	0.998905i	$-0.514894\pi$
$859$	23.2980	−	13.4511i	0.794916	−	0.458945i	0.841690	+	0.539960i	$0.181561\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	11.7980			0.401607			0.200804	−	0.979632i	$-0.435645\pi$
$863$	11.7980			0.401607			0.200804	+	0.979632i	$0.435645\pi$
$864$		0			0
$865$	−3.79796	+	2.19275i	−0.129134	+	0.0745558i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.494897	−	0.285729i	−0.0167882	−	0.00969270i
$870$		0			0
$871$	34.7474	+	20.0614i	1.17737	+	0.679756i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			54.7257i			1.85007i
$876$		0			0
$877$	−23.5454	−	13.5939i	−0.795072	−	0.459035i	0.0466731	−	0.998910i	$-0.485138\pi$
$877$	−23.5454	−	13.5939i	−0.795072	−	0.459035i	−0.841745	+	0.539875i	$0.818471\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−15.7980			−0.532247			−0.266123	−	0.963939i	$-0.585743\pi$
$881$	−15.7980			−0.532247			−0.266123	+	0.963939i	$0.585743\pi$
$882$		0			0
$883$	−38.2980	+	22.1113i	−1.28883	+	0.744106i	−0.978445	−	0.206506i	$-0.933791\pi$
$883$	−38.2980	+	22.1113i	−1.28883	+	0.744106i	−0.310384	+	0.950611i	$0.600457\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	26.5959	+	46.0655i	0.893004	+	1.54673i	0.836256	+	0.548339i	$0.184740\pi$
$887$	26.5959	+	46.0655i	0.893004	+	1.54673i	0.0567473	+	0.998389i	$0.481927\pi$
$888$		0			0
$889$	−22.8990	+	13.2207i	−0.768007	+	0.443409i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−39.4949	+	4.56048i	−1.32165	+	0.152611i
$894$		0			0
$895$	25.7980	−	44.6834i	0.862331	−	1.49360i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−15.0000	+	8.66025i	−0.500278	+	0.288836i
$900$		0			0
$901$		−	25.1701i		−	0.838538i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			33.9411i			1.12824i
$906$		0			0
$907$	−11.5959	+	20.0847i	−0.385036	+	0.666902i	−0.991774	−	0.128000i	$-0.959144\pi$
$907$	−11.5959	+	20.0847i	−0.385036	+	0.666902i	0.606738	+	0.794902i	$0.292478\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	15.5959			0.516716			0.258358	−	0.966049i	$-0.416819\pi$
$911$	15.5959			0.516716			0.258358	+	0.966049i	$0.416819\pi$
$912$		0			0
$913$	19.5959			0.648530
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−31.5959	+	54.7257i	−1.04339	+	1.80720i
$918$		0			0
$919$		−	25.9165i		−	0.854908i	−0.904037	−	0.427454i	$-0.859411\pi$
$919$		−	25.9165i		−	0.854908i	0.904037	−	0.427454i	$-0.140589\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	18.2419i		−	0.600440i
$924$		0			0
$925$	−0.949490	+	0.548188i	−0.0312190	+	0.0180243i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.89898	−	6.75323i	0.127921	−	0.221566i	−0.794950	−	0.606675i	$-0.792503\pi$
$929$	3.89898	−	6.75323i	0.127921	−	0.221566i	0.922871	+	0.385109i	$0.125836\pi$
$930$		0			0
$931$	55.1918	+	23.8988i	1.80884	+	0.783250i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−19.5959	+	11.3137i	−0.640855	+	0.369998i
$936$		0			0
$937$	14.5000	+	25.1147i	0.473694	+	0.820463i	0.999546	−	0.0301133i	$-0.00958681\pi$
$937$	14.5000	+	25.1147i	0.473694	+	0.820463i	−0.525852	+	0.850576i	$0.676253\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−9.79796	+	5.65685i	−0.319404	+	0.184408i	−0.651127	−	0.758969i	$-0.725703\pi$
$941$	−9.79796	+	5.65685i	−0.319404	+	0.184408i	0.331723	+	0.943377i	$0.392370\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−42.0000	−	24.2487i	−1.36482	−	0.787977i	−0.374556	−	0.927204i	$-0.622205\pi$
$947$	−42.0000	−	24.2487i	−1.36482	−	0.787977i	−0.990260	+	0.139227i	$0.955538\pi$
$948$		0			0
$949$		−	31.0019i		−	1.00637i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.10102	+	0.635674i	0.0356656	+	0.0205915i	0.517727	−	0.855546i	$-0.326778\pi$
$953$	1.10102	+	0.635674i	0.0356656	+	0.0205915i	−0.482061	+	0.876138i	$0.660112\pi$
$954$		0			0
$955$	42.0000	+	24.2487i	1.35909	+	0.784670i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−47.3939	+	27.3629i	−1.53043	+	0.883593i
$960$		0			0
$961$	31.3939			1.01271
$962$		0			0
$963$		0			0
$964$		0			0
$965$	21.0000	−	12.1244i	0.676014	−	0.390297i
$966$		0			0
$967$	21.3434	+	12.3226i	0.686356	+	0.396268i	0.802246	−	0.596994i	$-0.203638\pi$
$967$	21.3434	+	12.3226i	0.686356	+	0.396268i	−0.115889	+	0.993262i	$0.536972\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	26.5959	−	46.0655i	0.853504	−	1.47831i	−0.0245227	−	0.999699i	$-0.507807\pi$
$971$	26.5959	−	46.0655i	0.853504	−	1.47831i	0.878026	−	0.478612i	$-0.158860\pi$
$972$		0			0
$973$	8.94949	+	15.5010i	0.286907	+	0.496938i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	33.0197i		−	1.05639i	−0.849122	−	0.528197i	$-0.822868\pi$
$977$		−	33.0197i		−	1.05639i	0.849122	−	0.528197i	$-0.177132\pi$
$978$		0			0
$979$	−6.20204	−	10.7423i	−0.198218	−	0.343324i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.8990	−	32.7340i	0.602784	−	1.04405i	−0.389614	−	0.920978i	$-0.627391\pi$
$983$	18.8990	−	32.7340i	0.602784	−	1.04405i	0.992398	−	0.123074i	$-0.0392752\pi$
$984$		0			0
$985$	−2.00000	+	3.46410i	−0.0637253	+	0.110375i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	29.3939			0.934671
$990$		0			0
$991$	−2.84847	+	4.93369i	−0.0904846	+	0.156724i	−0.907715	−	0.419587i	$-0.862175\pi$
$991$	−2.84847	+	4.93369i	−0.0904846	+	0.156724i	0.817231	+	0.576311i	$0.195508\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			25.5201i			0.809040i
$996$		0			0
$997$	−7.05051	−	12.2118i	−0.223292	−	0.386753i	0.732514	−	0.680752i	$-0.238347\pi$
$997$	−7.05051	−	12.2118i	−0.223292	−	0.386753i	−0.955806	+	0.293999i	$0.905014\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.bm.i.559.1		4
3.2	odd	2		912.2.bb.d.559.1	yes	4
4.3	odd	2		2736.2.bm.j.559.2		4
12.11	even	2		912.2.bb.c.559.2	yes	4
19.12	odd	6		2736.2.bm.j.1855.1		4
57.50	even	6		912.2.bb.c.31.1	✓	4
76.31	even	6	inner	2736.2.bm.i.1855.2		4
228.107	odd	6		912.2.bb.d.31.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.bb.c.31.1	✓	4	57.50	even	6
912.2.bb.c.559.2	yes	4	12.11	even	2
912.2.bb.d.31.2	yes	4	228.107	odd	6
912.2.bb.d.559.1	yes	4	3.2	odd	2
2736.2.bm.i.559.1		4	1.1	even	1	trivial
2736.2.bm.i.1855.2		4	76.31	even	6	inner
2736.2.bm.j.559.2		4	4.3	odd	2
2736.2.bm.j.1855.1		4	19.12	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.bm (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{5} -4.56048i q^{7} +O(q^{10})\)	\(q+(-1.00000 + 1.73205i) q^{5} -4.56048i q^{7} -5.65685i q^{11} +(3.94949 - 2.28024i) q^{13} +(-1.00000 + 1.73205i) q^{17} +(-4.00000 - 1.73205i) q^{19} +(-4.89898 + 2.82843i) q^{23} +(0.500000 + 0.866025i) q^{25} +(1.89898 - 1.09638i) q^{29} -7.89898 q^{31} +(7.89898 + 4.56048i) q^{35} +1.09638i q^{37} +(-4.50000 - 2.59808i) q^{43} +(7.89898 - 4.56048i) q^{47} -13.7980 q^{49} +(-10.8990 + 6.29253i) q^{53} +(9.79796 + 5.65685i) q^{55} +(-5.89898 + 10.2173i) q^{59} +(4.94949 + 8.57277i) q^{61} +9.12096i q^{65} +(4.39898 + 7.61926i) q^{67} +(2.00000 - 3.46410i) q^{71} +(3.39898 - 5.88721i) q^{73} -25.7980 q^{77} +(0.0505103 - 0.0874863i) q^{79} +3.46410i q^{83} +(-2.00000 - 3.46410i) q^{85} +(1.89898 - 1.09638i) q^{89} +(-10.3990 - 18.0116i) q^{91} +(7.00000 - 5.19615i) q^{95} +(3.79796 + 2.19275i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5}+O(q^{10}) \) 4 * q - 4 * q^5	\( 4 q - 4 q^{5} + 6 q^{13} - 4 q^{17} - 16 q^{19} + 2 q^{25} - 12 q^{29} - 12 q^{31} + 12 q^{35} - 18 q^{43} + 12 q^{47} - 16 q^{49} - 24 q^{53} - 4 q^{59} + 10 q^{61} - 2 q^{67} + 8 q^{71} - 6 q^{73} - 64 q^{77} + 10 q^{79} - 8 q^{85} - 12 q^{89} - 22 q^{91} + 28 q^{95} - 24 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 + 6 * q^13 - 4 * q^17 - 16 * q^19 + 2 * q^25 - 12 * q^29 - 12 * q^31 + 12 * q^35 - 18 * q^43 + 12 * q^47 - 16 * q^49 - 24 * q^53 - 4 * q^59 + 10 * q^61 - 2 * q^67 + 8 * q^71 - 6 * q^73 - 64 * q^77 + 10 * q^79 - 8 * q^85 - 12 * q^89 - 22 * q^91 + 28 * q^95 - 24 * q^97

Introduction

L-functions

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