LMFDB - Embedded newform 4368.2.a.bh.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4368,2,Mod(1,4368)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4368.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,-1,0,-2,0,2,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$34.8786556029$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.27492$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.27492 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.27492 q^{11} -1.00000 q^{13} -4.27492 q^{15} +0.274917 q^{17} +2.27492 q^{19} -1.00000 q^{21} -2.27492 q^{23} +13.2749 q^{25} +1.00000 q^{27} +8.27492 q^{29} -8.00000 q^{31} +2.27492 q^{33} +4.27492 q^{35} -4.27492 q^{37} -1.00000 q^{39} +6.54983 q^{41} +2.27492 q^{43} -4.27492 q^{45} +1.00000 q^{49} +0.274917 q^{51} +10.0000 q^{53} -9.72508 q^{55} +2.27492 q^{57} -8.00000 q^{59} -12.2749 q^{61} -1.00000 q^{63} +4.27492 q^{65} -12.5498 q^{67} -2.27492 q^{69} -12.8248 q^{73} +13.2749 q^{75} -2.27492 q^{77} -12.5498 q^{79} +1.00000 q^{81} +4.54983 q^{83} -1.17525 q^{85} +8.27492 q^{87} -14.0000 q^{89} +1.00000 q^{91} -8.00000 q^{93} -9.72508 q^{95} +15.0997 q^{97} +2.27492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9} - 3 q^{11} - 2 q^{13} - q^{15} - 7 q^{17} - 3 q^{19} - 2 q^{21} + 3 q^{23} + 19 q^{25} + 2 q^{27} + 9 q^{29} - 16 q^{31} - 3 q^{33} + q^{35} - q^{37} - 2 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - q^5 - 2 * q^7 + 2 * q^9 - 3 * q^11 - 2 * q^13 - q^15 - 7 * q^17 - 3 * q^19 - 2 * q^21 + 3 * q^23 + 19 * q^25 + 2 * q^27 + 9 * q^29 - 16 * q^31 - 3 * q^33 + q^35 - q^37 - 2 * q^39 - 2 * q^41 - 3 * q^43 - q^45 + 2 * q^49 - 7 * q^51 + 20 * q^53 - 27 * q^55 - 3 * q^57 - 16 * q^59 - 17 * q^61 - 2 * q^63 + q^65 - 10 * q^67 + 3 * q^69 - 3 * q^73 + 19 * q^75 + 3 * q^77 - 10 * q^79 + 2 * q^81 - 6 * q^83 - 25 * q^85 + 9 * q^87 - 28 * q^89 + 2 * q^91 - 16 * q^93 - 27 * q^95 - 3 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−4.27492	−1.91180	−0.955901	−	0.293691i	$-0.905116\pi$
$5$	−4.27492	−1.91180	−0.955901	+	0.293691i	$0.905116\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.27492	0.685913	0.342957	−	0.939351i	$-0.388572\pi$
$11$	2.27492	0.685913	0.342957	+	0.939351i	$0.388572\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−4.27492	−1.10378
$16$	0	0
$17$	0.274917	0.0666772	0.0333386	−	0.999444i	$-0.489386\pi$
$17$	0.274917	0.0666772	0.0333386	+	0.999444i	$0.489386\pi$
$18$	0	0
$19$	2.27492	0.521902	0.260951	−	0.965352i	$-0.415964\pi$
$19$	2.27492	0.521902	0.260951	+	0.965352i	$0.415964\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−2.27492	−0.474353	−0.237177	−	0.971467i	$-0.576222\pi$
$23$	−2.27492	−0.474353	−0.237177	+	0.971467i	$0.576222\pi$
$24$	0	0
$25$	13.2749	2.65498
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.27492	1.53661	0.768307	−	0.640082i	$-0.221100\pi$
$29$	8.27492	1.53661	0.768307	+	0.640082i	$0.221100\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	2.27492	0.396012
$34$	0	0
$35$	4.27492	0.722593
$36$	0	0
$37$	−4.27492	−0.702792	−0.351396	−	0.936227i	$-0.614293\pi$
$37$	−4.27492	−0.702792	−0.351396	+	0.936227i	$0.614293\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	6.54983	1.02291	0.511456	−	0.859309i	$-0.329106\pi$
$41$	6.54983	1.02291	0.511456	+	0.859309i	$0.329106\pi$
$42$	0	0
$43$	2.27492	0.346922	0.173461	−	0.984841i	$-0.444505\pi$
$43$	2.27492	0.346922	0.173461	+	0.984841i	$0.444505\pi$
$44$	0	0
$45$	−4.27492	−0.637267
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.274917	0.0384961
$52$	0	0
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	−9.72508	−1.31133
$56$	0	0
$57$	2.27492	0.301320
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−12.2749	−1.57164	−0.785821	−	0.618454i	$-0.787759\pi$
$61$	−12.2749	−1.57164	−0.785821	+	0.618454i	$0.787759\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	4.27492	0.530238
$66$	0	0
$67$	−12.5498	−1.53321	−0.766603	−	0.642121i	$-0.778055\pi$
$67$	−12.5498	−1.53321	−0.766603	+	0.642121i	$0.778055\pi$
$68$	0	0
$69$	−2.27492	−0.273868
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−12.8248	−1.50102	−0.750512	−	0.660857i	$-0.770193\pi$
$73$	−12.8248	−1.50102	−0.750512	+	0.660857i	$0.770193\pi$
$74$	0	0
$75$	13.2749	1.53286
$76$	0	0
$77$	−2.27492	−0.259251
$78$	0	0
$79$	−12.5498	−1.41197	−0.705983	−	0.708228i	$-0.749495\pi$
$79$	−12.5498	−1.41197	−0.705983	+	0.708228i	$0.749495\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.54983	0.499409	0.249705	−	0.968322i	$-0.419666\pi$
$83$	4.54983	0.499409	0.249705	+	0.968322i	$0.419666\pi$
$84$	0	0
$85$	−1.17525	−0.127474
$86$	0	0
$87$	8.27492	0.887164
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	−9.72508	−0.997772
$96$	0	0
$97$	15.0997	1.53314	0.766570	−	0.642161i	$-0.221962\pi$
$97$	15.0997	1.53314	0.766570	+	0.642161i	$0.221962\pi$
$98$	0	0
$99$	2.27492	0.228638
$100$	0	0
$101$	2.54983	0.253718	0.126859	−	0.991921i	$-0.459510\pi$
$101$	2.54983	0.253718	0.126859	+	0.991921i	$0.459510\pi$
$102$	0	0
$103$	−10.2749	−1.01242	−0.506209	−	0.862411i	$-0.668954\pi$
$103$	−10.2749	−1.01242	−0.506209	+	0.862411i	$0.668954\pi$
$104$	0	0
$105$	4.27492	0.417189
$106$	0	0
$107$	−0.549834	−0.0531545	−0.0265773	−	0.999647i	$-0.508461\pi$
$107$	−0.549834	−0.0531545	−0.0265773	+	0.999647i	$0.508461\pi$
$108$	0	0
$109$	0.274917	0.0263323	0.0131661	−	0.999913i	$-0.495809\pi$
$109$	0.274917	0.0263323	0.0131661	+	0.999913i	$0.495809\pi$
$110$	0	0
$111$	−4.27492	−0.405757
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	9.72508	0.906869
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−0.274917	−0.0252016
$120$	0	0
$121$	−5.82475	−0.529523
$122$	0	0
$123$	6.54983	0.590579
$124$	0	0
$125$	−35.3746	−3.16400
$126$	0	0
$127$	20.5498	1.82350	0.911751	−	0.410742i	$-0.134730\pi$
$127$	20.5498	1.82350	0.911751	+	0.410742i	$0.134730\pi$
$128$	0	0
$129$	2.27492	0.200295
$130$	0	0
$131$	−6.82475	−0.596281	−0.298141	−	0.954522i	$-0.596366\pi$
$131$	−6.82475	−0.596281	−0.298141	+	0.954522i	$0.596366\pi$
$132$	0	0
$133$	−2.27492	−0.197260
$134$	0	0
$135$	−4.27492	−0.367926
$136$	0	0
$137$	−17.3746	−1.48441	−0.742206	−	0.670172i	$-0.766220\pi$
$137$	−17.3746	−1.48441	−0.742206	+	0.670172i	$0.766220\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.27492	−0.190238
$144$	0	0
$145$	−35.3746	−2.93770
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	22.5498	1.84735	0.923677	−	0.383172i	$-0.125168\pi$
$149$	22.5498	1.84735	0.923677	+	0.383172i	$0.125168\pi$
$150$	0	0
$151$	−6.27492	−0.510646	−0.255323	−	0.966856i	$-0.582182\pi$
$151$	−6.27492	−0.510646	−0.255323	+	0.966856i	$0.582182\pi$
$152$	0	0
$153$	0.274917	0.0222257
$154$	0	0
$155$	34.1993	2.74696
$156$	0	0
$157$	−13.3746	−1.06741	−0.533704	−	0.845671i	$-0.679200\pi$
$157$	−13.3746	−1.06741	−0.533704	+	0.845671i	$0.679200\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	2.27492	0.179289
$162$	0	0
$163$	−12.5498	−0.982979	−0.491489	−	0.870884i	$-0.663547\pi$
$163$	−12.5498	−0.982979	−0.491489	+	0.870884i	$0.663547\pi$
$164$	0	0
$165$	−9.72508	−0.757097
$166$	0	0
$167$	1.72508	0.133491	0.0667455	−	0.997770i	$-0.478738\pi$
$167$	1.72508	0.133491	0.0667455	+	0.997770i	$0.478738\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.27492	0.173967
$172$	0	0
$173$	7.09967	0.539778	0.269889	−	0.962891i	$-0.413013\pi$
$173$	7.09967	0.539778	0.269889	+	0.962891i	$0.413013\pi$
$174$	0	0
$175$	−13.2749	−1.00349
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	19.0997	1.41967	0.709834	−	0.704369i	$-0.248770\pi$
$181$	19.0997	1.41967	0.709834	+	0.704369i	$0.248770\pi$
$182$	0	0
$183$	−12.2749	−0.907388
$184$	0	0
$185$	18.2749	1.34360
$186$	0	0
$187$	0.625414	0.0457348
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−2.27492	−0.164607	−0.0823036	−	0.996607i	$-0.526228\pi$
$191$	−2.27492	−0.164607	−0.0823036	+	0.996607i	$0.526228\pi$
$192$	0	0
$193$	−18.5498	−1.33525	−0.667623	−	0.744499i	$-0.732688\pi$
$193$	−18.5498	−1.33525	−0.667623	+	0.744499i	$0.732688\pi$
$194$	0	0
$195$	4.27492	0.306133
$196$	0	0
$197$	−2.54983	−0.181668	−0.0908341	−	0.995866i	$-0.528953\pi$
$197$	−2.54983	−0.181668	−0.0908341	+	0.995866i	$0.528953\pi$
$198$	0	0
$199$	−6.82475	−0.483794	−0.241897	−	0.970302i	$-0.577770\pi$
$199$	−6.82475	−0.483794	−0.241897	+	0.970302i	$0.577770\pi$
$200$	0	0
$201$	−12.5498	−0.885197
$202$	0	0
$203$	−8.27492	−0.580785
$204$	0	0
$205$	−28.0000	−1.95560
$206$	0	0
$207$	−2.27492	−0.158118
$208$	0	0
$209$	5.17525	0.357979
$210$	0	0
$211$	−1.17525	−0.0809074	−0.0404537	−	0.999181i	$-0.512880\pi$
$211$	−1.17525	−0.0809074	−0.0404537	+	0.999181i	$0.512880\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9.72508	−0.663245
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	−12.8248	−0.866616
$220$	0	0
$221$	−0.274917	−0.0184929
$222$	0	0
$223$	−21.6495	−1.44976	−0.724879	−	0.688876i	$-0.758104\pi$
$223$	−21.6495	−1.44976	−0.724879	+	0.688876i	$0.758104\pi$
$224$	0	0
$225$	13.2749	0.884994
$226$	0	0
$227$	20.5498	1.36394	0.681970	−	0.731380i	$-0.261123\pi$
$227$	20.5498	1.36394	0.681970	+	0.731380i	$0.261123\pi$
$228$	0	0
$229$	19.0997	1.26214	0.631071	−	0.775725i	$-0.282616\pi$
$229$	19.0997	1.26214	0.631071	+	0.775725i	$0.282616\pi$
$230$	0	0
$231$	−2.27492	−0.149679
$232$	0	0
$233$	5.45017	0.357052	0.178526	−	0.983935i	$-0.442867\pi$
$233$	5.45017	0.357052	0.178526	+	0.983935i	$0.442867\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−12.5498	−0.815199
$238$	0	0
$239$	−25.0997	−1.62356	−0.811781	−	0.583962i	$-0.801502\pi$
$239$	−25.0997	−1.62356	−0.811781	+	0.583962i	$0.801502\pi$
$240$	0	0
$241$	15.0997	0.972655	0.486328	−	0.873777i	$-0.338336\pi$
$241$	15.0997	0.972655	0.486328	+	0.873777i	$0.338336\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.27492	−0.273114
$246$	0	0
$247$	−2.27492	−0.144750
$248$	0	0
$249$	4.54983	0.288334
$250$	0	0
$251$	−23.9244	−1.51010	−0.755048	−	0.655669i	$-0.772386\pi$
$251$	−23.9244	−1.51010	−0.755048	+	0.655669i	$0.772386\pi$
$252$	0	0
$253$	−5.17525	−0.325365
$254$	0	0
$255$	−1.17525	−0.0735969
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	4.27492	0.265630
$260$	0	0
$261$	8.27492	0.512205
$262$	0	0
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	0	0
$265$	−42.7492	−2.62606
$266$	0	0
$267$	−14.0000	−0.856786
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−28.5498	−1.73428	−0.867139	−	0.498065i	$-0.834044\pi$
$271$	−28.5498	−1.73428	−0.867139	+	0.498065i	$0.834044\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	30.1993	1.82109
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	15.0997	0.900771	0.450385	−	0.892834i	$-0.351287\pi$
$281$	15.0997	0.900771	0.450385	+	0.892834i	$0.351287\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	−9.72508	−0.576064
$286$	0	0
$287$	−6.54983	−0.386625
$288$	0	0
$289$	−16.9244	−0.995554
$290$	0	0
$291$	15.0997	0.885158
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	34.1993	1.99116
$296$	0	0
$297$	2.27492	0.132004
$298$	0	0
$299$	2.27492	0.131562
$300$	0	0
$301$	−2.27492	−0.131124
$302$	0	0
$303$	2.54983	0.146484
$304$	0	0
$305$	52.4743	3.00467
$306$	0	0
$307$	−21.0997	−1.20422	−0.602111	−	0.798412i	$-0.705673\pi$
$307$	−21.0997	−1.20422	−0.602111	+	0.798412i	$0.705673\pi$
$308$	0	0
$309$	−10.2749	−0.584520
$310$	0	0
$311$	−3.45017	−0.195641	−0.0978205	−	0.995204i	$-0.531187\pi$
$311$	−3.45017	−0.195641	−0.0978205	+	0.995204i	$0.531187\pi$
$312$	0	0
$313$	−27.6495	−1.56284	−0.781421	−	0.624004i	$-0.785505\pi$
$313$	−27.6495	−1.56284	−0.781421	+	0.624004i	$0.785505\pi$
$314$	0	0
$315$	4.27492	0.240864
$316$	0	0
$317$	−1.45017	−0.0814494	−0.0407247	−	0.999170i	$-0.512967\pi$
$317$	−1.45017	−0.0814494	−0.0407247	+	0.999170i	$0.512967\pi$
$318$	0	0
$319$	18.8248	1.05398
$320$	0	0
$321$	−0.549834	−0.0306888
$322$	0	0
$323$	0.625414	0.0347990
$324$	0	0
$325$	−13.2749	−0.736360
$326$	0	0
$327$	0.274917	0.0152030
$328$	0	0
$329$	0	0
$330$	0	0
$331$	29.6495	1.62968	0.814842	−	0.579683i	$-0.196824\pi$
$331$	29.6495	1.62968	0.814842	+	0.579683i	$0.196824\pi$
$332$	0	0
$333$	−4.27492	−0.234264
$334$	0	0
$335$	53.6495	2.93119
$336$	0	0
$337$	9.37459	0.510666	0.255333	−	0.966853i	$-0.417815\pi$
$337$	9.37459	0.510666	0.255333	+	0.966853i	$0.417815\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−18.1993	−0.985549
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	9.72508	0.523581
$346$	0	0
$347$	−5.09967	−0.273765	−0.136882	−	0.990587i	$-0.543708\pi$
$347$	−5.09967	−0.273765	−0.136882	+	0.990587i	$0.543708\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	27.0997	1.44237	0.721185	−	0.692743i	$-0.243598\pi$
$353$	27.0997	1.44237	0.721185	+	0.692743i	$0.243598\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.274917	−0.0145502
$358$	0	0
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	−13.8248	−0.727619
$362$	0	0
$363$	−5.82475	−0.305720
$364$	0	0
$365$	54.8248	2.86966
$366$	0	0
$367$	2.90033	0.151396	0.0756980	−	0.997131i	$-0.475881\pi$
$367$	2.90033	0.151396	0.0756980	+	0.997131i	$0.475881\pi$
$368$	0	0
$369$	6.54983	0.340971
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	26.5498	1.37470	0.687349	−	0.726327i	$-0.258774\pi$
$373$	26.5498	1.37470	0.687349	+	0.726327i	$0.258774\pi$
$374$	0	0
$375$	−35.3746	−1.82674
$376$	0	0
$377$	−8.27492	−0.426180
$378$	0	0
$379$	33.0997	1.70022	0.850108	−	0.526609i	$-0.176537\pi$
$379$	33.0997	1.70022	0.850108	+	0.526609i	$0.176537\pi$
$380$	0	0
$381$	20.5498	1.05280
$382$	0	0
$383$	−30.2749	−1.54698	−0.773488	−	0.633811i	$-0.781490\pi$
$383$	−30.2749	−1.54698	−0.773488	+	0.633811i	$0.781490\pi$
$384$	0	0
$385$	9.72508	0.495636
$386$	0	0
$387$	2.27492	0.115641
$388$	0	0
$389$	28.1993	1.42976	0.714882	−	0.699246i	$-0.246481\pi$
$389$	28.1993	1.42976	0.714882	+	0.699246i	$0.246481\pi$
$390$	0	0
$391$	−0.625414	−0.0316285
$392$	0	0
$393$	−6.82475	−0.344263
$394$	0	0
$395$	53.6495	2.69940
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	−2.27492	−0.113888
$400$	0	0
$401$	−3.09967	−0.154790	−0.0773950	−	0.997001i	$-0.524660\pi$
$401$	−3.09967	−0.154790	−0.0773950	+	0.997001i	$0.524660\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	0	0
$405$	−4.27492	−0.212422
$406$	0	0
$407$	−9.72508	−0.482054
$408$	0	0
$409$	−7.17525	−0.354793	−0.177397	−	0.984139i	$-0.556768\pi$
$409$	−7.17525	−0.354793	−0.177397	+	0.984139i	$0.556768\pi$
$410$	0	0
$411$	−17.3746	−0.857025
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	−19.4502	−0.954771
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−2.27492	−0.111137	−0.0555685	−	0.998455i	$-0.517697\pi$
$419$	−2.27492	−0.111137	−0.0555685	+	0.998455i	$0.517697\pi$
$420$	0	0
$421$	3.09967	0.151069	0.0755343	−	0.997143i	$-0.475934\pi$
$421$	3.09967	0.151069	0.0755343	+	0.997143i	$0.475934\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.64950	0.177027
$426$	0	0
$427$	12.2749	0.594025
$428$	0	0
$429$	−2.27492	−0.109834
$430$	0	0
$431$	11.4502	0.551535	0.275768	−	0.961224i	$-0.411068\pi$
$431$	11.4502	0.551535	0.275768	+	0.961224i	$0.411068\pi$
$432$	0	0
$433$	23.6495	1.13652	0.568261	−	0.822848i	$-0.307616\pi$
$433$	23.6495	1.13652	0.568261	+	0.822848i	$0.307616\pi$
$434$	0	0
$435$	−35.3746	−1.69608
$436$	0	0
$437$	−5.17525	−0.247566
$438$	0	0
$439$	14.8248	0.707547	0.353773	−	0.935331i	$-0.384898\pi$
$439$	14.8248	0.707547	0.353773	+	0.935331i	$0.384898\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−7.45017	−0.353968	−0.176984	−	0.984214i	$-0.556634\pi$
$443$	−7.45017	−0.353968	−0.176984	+	0.984214i	$0.556634\pi$
$444$	0	0
$445$	59.8488	2.83711
$446$	0	0
$447$	22.5498	1.06657
$448$	0	0
$449$	−0.274917	−0.0129741	−0.00648707	−	0.999979i	$-0.502065\pi$
$449$	−0.274917	−0.0129741	−0.00648707	+	0.999979i	$0.502065\pi$
$450$	0	0
$451$	14.9003	0.701629
$452$	0	0
$453$	−6.27492	−0.294821
$454$	0	0
$455$	−4.27492	−0.200411
$456$	0	0
$457$	−18.5498	−0.867725	−0.433862	−	0.900979i	$-0.642850\pi$
$457$	−18.5498	−0.867725	−0.433862	+	0.900979i	$0.642850\pi$
$458$	0	0
$459$	0.274917	0.0128320
$460$	0	0
$461$	−17.9244	−0.834823	−0.417412	−	0.908717i	$-0.637063\pi$
$461$	−17.9244	−0.834823	−0.417412	+	0.908717i	$0.637063\pi$
$462$	0	0
$463$	−30.2749	−1.40699	−0.703497	−	0.710698i	$-0.748379\pi$
$463$	−30.2749	−1.40699	−0.703497	+	0.710698i	$0.748379\pi$
$464$	0	0
$465$	34.1993	1.58596
$466$	0	0
$467$	26.2749	1.21586	0.607929	−	0.793991i	$-0.292000\pi$
$467$	26.2749	1.21586	0.607929	+	0.793991i	$0.292000\pi$
$468$	0	0
$469$	12.5498	0.579498
$470$	0	0
$471$	−13.3746	−0.616268
$472$	0	0
$473$	5.17525	0.237958
$474$	0	0
$475$	30.1993	1.38564
$476$	0	0
$477$	10.0000	0.457869
$478$	0	0
$479$	23.3746	1.06801	0.534006	−	0.845481i	$-0.320686\pi$
$479$	23.3746	1.06801	0.534006	+	0.845481i	$0.320686\pi$
$480$	0	0
$481$	4.27492	0.194919
$482$	0	0
$483$	2.27492	0.103512
$484$	0	0
$485$	−64.5498	−2.93106
$486$	0	0
$487$	18.1993	0.824691	0.412345	−	0.911028i	$-0.364710\pi$
$487$	18.1993	0.824691	0.412345	+	0.911028i	$0.364710\pi$
$488$	0	0
$489$	−12.5498	−0.567523
$490$	0	0
$491$	−8.54983	−0.385849	−0.192924	−	0.981214i	$-0.561797\pi$
$491$	−8.54983	−0.385849	−0.192924	+	0.981214i	$0.561797\pi$
$492$	0	0
$493$	2.27492	0.102457
$494$	0	0
$495$	−9.72508	−0.437110
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−25.0997	−1.12362	−0.561808	−	0.827268i	$-0.689894\pi$
$499$	−25.0997	−1.12362	−0.561808	+	0.827268i	$0.689894\pi$
$500$	0	0
$501$	1.72508	0.0770710
$502$	0	0
$503$	3.45017	0.153835	0.0769176	−	0.997037i	$-0.475492\pi$
$503$	3.45017	0.153835	0.0769176	+	0.997037i	$0.475492\pi$
$504$	0	0
$505$	−10.9003	−0.485058
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−9.92442	−0.439892	−0.219946	−	0.975512i	$-0.570588\pi$
$509$	−9.92442	−0.439892	−0.219946	+	0.975512i	$0.570588\pi$
$510$	0	0
$511$	12.8248	0.567334
$512$	0	0
$513$	2.27492	0.100440
$514$	0	0
$515$	43.9244	1.93554
$516$	0	0
$517$	0	0
$518$	0	0
$519$	7.09967	0.311641
$520$	0	0
$521$	−4.27492	−0.187288	−0.0936438	−	0.995606i	$-0.529851\pi$
$521$	−4.27492	−0.187288	−0.0936438	+	0.995606i	$0.529851\pi$
$522$	0	0
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	0	0
$525$	−13.2749	−0.579365
$526$	0	0
$527$	−2.19934	−0.0958047
$528$	0	0
$529$	−17.8248	−0.774989
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	−6.54983	−0.283705
$534$	0	0
$535$	2.35050	0.101621
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	2.27492	0.0979876
$540$	0	0
$541$	42.4743	1.82611	0.913055	−	0.407835i	$-0.133716\pi$
$541$	42.4743	1.82611	0.913055	+	0.407835i	$0.133716\pi$
$542$	0	0
$543$	19.0997	0.819645
$544$	0	0
$545$	−1.17525	−0.0503421
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	−12.2749	−0.523881
$550$	0	0
$551$	18.8248	0.801961
$552$	0	0
$553$	12.5498	0.533673
$554$	0	0
$555$	18.2749	0.775727
$556$	0	0
$557$	−44.7492	−1.89608	−0.948042	−	0.318146i	$-0.896940\pi$
$557$	−44.7492	−1.89608	−0.948042	+	0.318146i	$0.896940\pi$
$558$	0	0
$559$	−2.27492	−0.0962187
$560$	0	0
$561$	0.625414	0.0264050
$562$	0	0
$563$	35.3746	1.49086	0.745431	−	0.666583i	$-0.232244\pi$
$563$	35.3746	1.49086	0.745431	+	0.666583i	$0.232244\pi$
$564$	0	0
$565$	25.6495	1.07908
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	3.09967	0.129945	0.0649724	−	0.997887i	$-0.479304\pi$
$569$	3.09967	0.129945	0.0649724	+	0.997887i	$0.479304\pi$
$570$	0	0
$571$	37.0997	1.55257	0.776286	−	0.630380i	$-0.217101\pi$
$571$	37.0997	1.55257	0.776286	+	0.630380i	$0.217101\pi$
$572$	0	0
$573$	−2.27492	−0.0950360
$574$	0	0
$575$	−30.1993	−1.25940
$576$	0	0
$577$	−8.90033	−0.370526	−0.185263	−	0.982689i	$-0.559314\pi$
$577$	−8.90033	−0.370526	−0.185263	+	0.982689i	$0.559314\pi$
$578$	0	0
$579$	−18.5498	−0.770905
$580$	0	0
$581$	−4.54983	−0.188759
$582$	0	0
$583$	22.7492	0.942174
$584$	0	0
$585$	4.27492	0.176746
$586$	0	0
$587$	−46.7492	−1.92954	−0.964772	−	0.263086i	$-0.915260\pi$
$587$	−46.7492	−1.92954	−0.964772	+	0.263086i	$0.915260\pi$
$588$	0	0
$589$	−18.1993	−0.749891
$590$	0	0
$591$	−2.54983	−0.104886
$592$	0	0
$593$	−7.09967	−0.291548	−0.145774	−	0.989318i	$-0.546567\pi$
$593$	−7.09967	−0.291548	−0.145774	+	0.989318i	$0.546567\pi$
$594$	0	0
$595$	1.17525	0.0481805
$596$	0	0
$597$	−6.82475	−0.279318
$598$	0	0
$599$	−21.7251	−0.887663	−0.443831	−	0.896110i	$-0.646381\pi$
$599$	−21.7251	−0.887663	−0.443831	+	0.896110i	$0.646381\pi$
$600$	0	0
$601$	23.6495	0.964683	0.482342	−	0.875983i	$-0.339786\pi$
$601$	23.6495	0.964683	0.482342	+	0.875983i	$0.339786\pi$
$602$	0	0
$603$	−12.5498	−0.511069
$604$	0	0
$605$	24.9003	1.01234
$606$	0	0
$607$	−9.17525	−0.372412	−0.186206	−	0.982511i	$-0.559619\pi$
$607$	−9.17525	−0.372412	−0.186206	+	0.982511i	$0.559619\pi$
$608$	0	0
$609$	−8.27492	−0.335317
$610$	0	0
$611$	0	0
$612$	0	0
$613$	16.2749	0.657338	0.328669	−	0.944445i	$-0.393400\pi$
$613$	16.2749	0.657338	0.328669	+	0.944445i	$0.393400\pi$
$614$	0	0
$615$	−28.0000	−1.12907
$616$	0	0
$617$	−20.8248	−0.838373	−0.419186	−	0.907900i	$-0.637685\pi$
$617$	−20.8248	−0.838373	−0.419186	+	0.907900i	$0.637685\pi$
$618$	0	0
$619$	−29.7251	−1.19475	−0.597376	−	0.801961i	$-0.703790\pi$
$619$	−29.7251	−1.19475	−0.597376	+	0.801961i	$0.703790\pi$
$620$	0	0
$621$	−2.27492	−0.0912893
$622$	0	0
$623$	14.0000	0.560898
$624$	0	0
$625$	84.8488	3.39395
$626$	0	0
$627$	5.17525	0.206680
$628$	0	0
$629$	−1.17525	−0.0468602
$630$	0	0
$631$	10.8248	0.430927	0.215463	−	0.976512i	$-0.430874\pi$
$631$	10.8248	0.430927	0.215463	+	0.976512i	$0.430874\pi$
$632$	0	0
$633$	−1.17525	−0.0467119
$634$	0	0
$635$	−87.8488	−3.48617
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−15.0997	−0.596401	−0.298201	−	0.954503i	$-0.596386\pi$
$641$	−15.0997	−0.596401	−0.298201	+	0.954503i	$0.596386\pi$
$642$	0	0
$643$	23.9244	0.943487	0.471744	−	0.881736i	$-0.343625\pi$
$643$	23.9244	0.943487	0.471744	+	0.881736i	$0.343625\pi$
$644$	0	0
$645$	−9.72508	−0.382925
$646$	0	0
$647$	18.1993	0.715490	0.357745	−	0.933819i	$-0.383546\pi$
$647$	18.1993	0.715490	0.357745	+	0.933819i	$0.383546\pi$
$648$	0	0
$649$	−18.1993	−0.714386
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	−5.37459	−0.210324	−0.105162	−	0.994455i	$-0.533536\pi$
$653$	−5.37459	−0.210324	−0.105162	+	0.994455i	$0.533536\pi$
$654$	0	0
$655$	29.1752	1.13997
$656$	0	0
$657$	−12.8248	−0.500341
$658$	0	0
$659$	7.45017	0.290217	0.145109	−	0.989416i	$-0.453647\pi$
$659$	7.45017	0.290217	0.145109	+	0.989416i	$0.453647\pi$
$660$	0	0
$661$	−40.1993	−1.56357	−0.781787	−	0.623546i	$-0.785691\pi$
$661$	−40.1993	−1.56357	−0.781787	+	0.623546i	$0.785691\pi$
$662$	0	0
$663$	−0.274917	−0.0106769
$664$	0	0
$665$	9.72508	0.377123
$666$	0	0
$667$	−18.8248	−0.728897
$668$	0	0
$669$	−21.6495	−0.837018
$670$	0	0
$671$	−27.9244	−1.07801
$672$	0	0
$673$	16.2749	0.627352	0.313676	−	0.949530i	$-0.398439\pi$
$673$	16.2749	0.627352	0.313676	+	0.949530i	$0.398439\pi$
$674$	0	0
$675$	13.2749	0.510952
$676$	0	0
$677$	16.1993	0.622591	0.311296	−	0.950313i	$-0.399237\pi$
$677$	16.1993	0.622591	0.311296	+	0.950313i	$0.399237\pi$
$678$	0	0
$679$	−15.0997	−0.579472
$680$	0	0
$681$	20.5498	0.787471
$682$	0	0
$683$	−3.37459	−0.129125	−0.0645625	−	0.997914i	$-0.520565\pi$
$683$	−3.37459	−0.129125	−0.0645625	+	0.997914i	$0.520565\pi$
$684$	0	0
$685$	74.2749	2.83790
$686$	0	0
$687$	19.0997	0.728698
$688$	0	0
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	0	0
$693$	−2.27492	−0.0864170
$694$	0	0
$695$	17.0997	0.648627
$696$	0	0
$697$	1.80066	0.0682049
$698$	0	0
$699$	5.45017	0.206144
$700$	0	0
$701$	−15.0997	−0.570307	−0.285153	−	0.958482i	$-0.592045\pi$
$701$	−15.0997	−0.570307	−0.285153	+	0.958482i	$0.592045\pi$
$702$	0	0
$703$	−9.72508	−0.366788
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.54983	−0.0958964
$708$	0	0
$709$	−23.0997	−0.867526	−0.433763	−	0.901027i	$-0.642815\pi$
$709$	−23.0997	−0.867526	−0.433763	+	0.901027i	$0.642815\pi$
$710$	0	0
$711$	−12.5498	−0.470656
$712$	0	0
$713$	18.1993	0.681571
$714$	0	0
$715$	9.72508	0.363697
$716$	0	0
$717$	−25.0997	−0.937364
$718$	0	0
$719$	29.6495	1.10574	0.552870	−	0.833268i	$-0.313533\pi$
$719$	29.6495	1.10574	0.552870	+	0.833268i	$0.313533\pi$
$720$	0	0
$721$	10.2749	0.382658
$722$	0	0
$723$	15.0997	0.561563
$724$	0	0
$725$	109.849	4.07968
$726$	0	0
$727$	−35.3746	−1.31197	−0.655985	−	0.754774i	$-0.727747\pi$
$727$	−35.3746	−1.31197	−0.655985	+	0.754774i	$0.727747\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.625414	0.0231318
$732$	0	0
$733$	14.5498	0.537410	0.268705	−	0.963222i	$-0.413404\pi$
$733$	14.5498	0.537410	0.268705	+	0.963222i	$0.413404\pi$
$734$	0	0
$735$	−4.27492	−0.157683
$736$	0	0
$737$	−28.5498	−1.05165
$738$	0	0
$739$	−37.6495	−1.38496	−0.692480	−	0.721437i	$-0.743482\pi$
$739$	−37.6495	−1.38496	−0.692480	+	0.721437i	$0.743482\pi$
$740$	0	0
$741$	−2.27492	−0.0835712
$742$	0	0
$743$	−11.4502	−0.420066	−0.210033	−	0.977694i	$-0.567357\pi$
$743$	−11.4502	−0.420066	−0.210033	+	0.977694i	$0.567357\pi$
$744$	0	0
$745$	−96.3987	−3.53177
$746$	0	0
$747$	4.54983	0.166470
$748$	0	0
$749$	0.549834	0.0200905
$750$	0	0
$751$	−10.1993	−0.372179	−0.186090	−	0.982533i	$-0.559581\pi$
$751$	−10.1993	−0.372179	−0.186090	+	0.982533i	$0.559581\pi$
$752$	0	0
$753$	−23.9244	−0.871854
$754$	0	0
$755$	26.8248	0.976253
$756$	0	0
$757$	−35.0997	−1.27572	−0.637860	−	0.770153i	$-0.720180\pi$
$757$	−35.0997	−1.27572	−0.637860	+	0.770153i	$0.720180\pi$
$758$	0	0
$759$	−5.17525	−0.187850
$760$	0	0
$761$	38.5498	1.39743	0.698715	−	0.715400i	$-0.253755\pi$
$761$	38.5498	1.39743	0.698715	+	0.715400i	$0.253755\pi$
$762$	0	0
$763$	−0.274917	−0.00995267
$764$	0	0
$765$	−1.17525	−0.0424912
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	32.8248	1.18369	0.591845	−	0.806051i	$-0.298400\pi$
$769$	32.8248	1.18369	0.591845	+	0.806051i	$0.298400\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	−9.92442	−0.356957	−0.178478	−	0.983944i	$-0.557117\pi$
$773$	−9.92442	−0.356957	−0.178478	+	0.983944i	$0.557117\pi$
$774$	0	0
$775$	−106.199	−3.81479
$776$	0	0
$777$	4.27492	0.153362
$778$	0	0
$779$	14.9003	0.533860
$780$	0	0
$781$	0	0
$782$	0	0
$783$	8.27492	0.295721
$784$	0	0
$785$	57.1752	2.04067
$786$	0	0
$787$	10.2749	0.366261	0.183131	−	0.983089i	$-0.441377\pi$
$787$	10.2749	0.366261	0.183131	+	0.983089i	$0.441377\pi$
$788$	0	0
$789$	−28.0000	−0.996826
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	12.2749	0.435895
$794$	0	0
$795$	−42.7492	−1.51616
$796$	0	0
$797$	−15.6495	−0.554334	−0.277167	−	0.960822i	$-0.589396\pi$
$797$	−15.6495	−0.554334	−0.277167	+	0.960822i	$0.589396\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	−29.1752	−1.02957
$804$	0	0
$805$	−9.72508	−0.342764
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	−56.1993	−1.97586	−0.987932	−	0.154890i	$-0.950498\pi$
$809$	−56.1993	−1.97586	−0.987932	+	0.154890i	$0.950498\pi$
$810$	0	0
$811$	11.3746	0.399416	0.199708	−	0.979855i	$-0.436001\pi$
$811$	11.3746	0.399416	0.199708	+	0.979855i	$0.436001\pi$
$812$	0	0
$813$	−28.5498	−1.00129
$814$	0	0
$815$	53.6495	1.87926
$816$	0	0
$817$	5.17525	0.181059
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	31.6495	1.10458	0.552288	−	0.833654i	$-0.313755\pi$
$821$	31.6495	1.10458	0.552288	+	0.833654i	$0.313755\pi$
$822$	0	0
$823$	25.0997	0.874919	0.437460	−	0.899238i	$-0.355878\pi$
$823$	25.0997	0.874919	0.437460	+	0.899238i	$0.355878\pi$
$824$	0	0
$825$	30.1993	1.05141
$826$	0	0
$827$	26.2749	0.913668	0.456834	−	0.889552i	$-0.348983\pi$
$827$	26.2749	0.913668	0.456834	+	0.889552i	$0.348983\pi$
$828$	0	0
$829$	11.7251	0.407229	0.203614	−	0.979051i	$-0.434731\pi$
$829$	11.7251	0.407229	0.203614	+	0.979051i	$0.434731\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	0.274917	0.00952532
$834$	0	0
$835$	−7.37459	−0.255208
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	−22.9003	−0.790607	−0.395304	−	0.918551i	$-0.629361\pi$
$839$	−22.9003	−0.790607	−0.395304	+	0.918551i	$0.629361\pi$
$840$	0	0
$841$	39.4743	1.36118
$842$	0	0
$843$	15.0997	0.520060
$844$	0	0
$845$	−4.27492	−0.147062
$846$	0	0
$847$	5.82475	0.200141
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	9.72508	0.333372
$852$	0	0
$853$	−27.6495	−0.946701	−0.473350	−	0.880874i	$-0.656956\pi$
$853$	−27.6495	−0.946701	−0.473350	+	0.880874i	$0.656956\pi$
$854$	0	0
$855$	−9.72508	−0.332591
$856$	0	0
$857$	19.0997	0.652432	0.326216	−	0.945295i	$-0.394226\pi$
$857$	19.0997	0.652432	0.326216	+	0.945295i	$0.394226\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	−6.54983	−0.223218
$862$	0	0
$863$	−29.6495	−1.00928	−0.504640	−	0.863330i	$-0.668375\pi$
$863$	−29.6495	−1.00928	−0.504640	+	0.863330i	$0.668375\pi$
$864$	0	0
$865$	−30.3505	−1.03195
$866$	0	0
$867$	−16.9244	−0.574783
$868$	0	0
$869$	−28.5498	−0.968487
$870$	0	0
$871$	12.5498	0.425235
$872$	0	0
$873$	15.0997	0.511046
$874$	0	0
$875$	35.3746	1.19588
$876$	0	0
$877$	51.0997	1.72551	0.862757	−	0.505619i	$-0.168736\pi$
$877$	51.0997	1.72551	0.862757	+	0.505619i	$0.168736\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−31.7251	−1.06885	−0.534423	−	0.845217i	$-0.679471\pi$
$881$	−31.7251	−1.06885	−0.534423	+	0.845217i	$0.679471\pi$
$882$	0	0
$883$	−31.9244	−1.07434	−0.537171	−	0.843473i	$-0.680507\pi$
$883$	−31.9244	−1.07434	−0.537171	+	0.843473i	$0.680507\pi$
$884$	0	0
$885$	34.1993	1.14960
$886$	0	0
$887$	33.0997	1.11138	0.555689	−	0.831390i	$-0.312455\pi$
$887$	33.0997	1.11138	0.555689	+	0.831390i	$0.312455\pi$
$888$	0	0
$889$	−20.5498	−0.689219
$890$	0	0
$891$	2.27492	0.0762126
$892$	0	0
$893$	0	0
$894$	0	0
$895$	51.2990	1.71474
$896$	0	0
$897$	2.27492	0.0759573
$898$	0	0
$899$	−66.1993	−2.20787
$900$	0	0
$901$	2.74917	0.0915882
$902$	0	0
$903$	−2.27492	−0.0757045
$904$	0	0
$905$	−81.6495	−2.71412
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	2.54983	0.0845727
$910$	0	0
$911$	52.4743	1.73855	0.869275	−	0.494329i	$-0.164586\pi$
$911$	52.4743	1.73855	0.869275	+	0.494329i	$0.164586\pi$
$912$	0	0
$913$	10.3505	0.342551
$914$	0	0
$915$	52.4743	1.73475
$916$	0	0
$917$	6.82475	0.225373
$918$	0	0
$919$	12.5498	0.413981	0.206990	−	0.978343i	$-0.433633\pi$
$919$	12.5498	0.413981	0.206990	+	0.978343i	$0.433633\pi$
$920$	0	0
$921$	−21.0997	−0.695258
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−56.7492	−1.86590
$926$	0	0
$927$	−10.2749	−0.337473
$928$	0	0
$929$	14.5498	0.477365	0.238682	−	0.971098i	$-0.423285\pi$
$929$	14.5498	0.477365	0.238682	+	0.971098i	$0.423285\pi$
$930$	0	0
$931$	2.27492	0.0745574
$932$	0	0
$933$	−3.45017	−0.112953
$934$	0	0
$935$	−2.67359	−0.0874358
$936$	0	0
$937$	16.9003	0.552110	0.276055	−	0.961142i	$-0.410973\pi$
$937$	16.9003	0.552110	0.276055	+	0.961142i	$0.410973\pi$
$938$	0	0
$939$	−27.6495	−0.902307
$940$	0	0
$941$	51.0997	1.66580	0.832901	−	0.553422i	$-0.186678\pi$
$941$	51.0997	1.66580	0.832901	+	0.553422i	$0.186678\pi$
$942$	0	0
$943$	−14.9003	−0.485222
$944$	0	0
$945$	4.27492	0.139063
$946$	0	0
$947$	17.1752	0.558121	0.279060	−	0.960274i	$-0.409977\pi$
$947$	17.1752	0.558121	0.279060	+	0.960274i	$0.409977\pi$
$948$	0	0
$949$	12.8248	0.416309
$950$	0	0
$951$	−1.45017	−0.0470248
$952$	0	0
$953$	31.6495	1.02523	0.512614	−	0.858619i	$-0.328677\pi$
$953$	31.6495	1.02523	0.512614	+	0.858619i	$0.328677\pi$
$954$	0	0
$955$	9.72508	0.314696
$956$	0	0
$957$	18.8248	0.608518
$958$	0	0
$959$	17.3746	0.561055
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−0.549834	−0.0177182
$964$	0	0
$965$	79.2990	2.55273
$966$	0	0
$967$	−42.8248	−1.37715	−0.688576	−	0.725165i	$-0.741764\pi$
$967$	−42.8248	−1.37715	−0.688576	+	0.725165i	$0.741764\pi$
$968$	0	0
$969$	0.625414	0.0200912
$970$	0	0
$971$	29.0997	0.933853	0.466926	−	0.884296i	$-0.345361\pi$
$971$	29.0997	0.933853	0.466926	+	0.884296i	$0.345361\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	0	0
$975$	−13.2749	−0.425138
$976$	0	0
$977$	−13.9244	−0.445482	−0.222741	−	0.974878i	$-0.571500\pi$
$977$	−13.9244	−0.445482	−0.222741	+	0.974878i	$0.571500\pi$
$978$	0	0
$979$	−31.8488	−1.01789
$980$	0	0
$981$	0.274917	0.00877743
$982$	0	0
$983$	−61.0241	−1.94637	−0.973183	−	0.230032i	$-0.926117\pi$
$983$	−61.0241	−1.94637	−0.973183	+	0.230032i	$0.926117\pi$
$984$	0	0
$985$	10.9003	0.347313
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.17525	−0.164563
$990$	0	0
$991$	46.7492	1.48504	0.742518	−	0.669826i	$-0.233631\pi$
$991$	46.7492	1.48504	0.742518	+	0.669826i	$0.233631\pi$
$992$	0	0
$993$	29.6495	0.940899
$994$	0	0
$995$	29.1752	0.924918
$996$	0	0
$997$	−12.9003	−0.408558	−0.204279	−	0.978913i	$-0.565485\pi$
$997$	−12.9003	−0.408558	−0.204279	+	0.978913i	$0.565485\pi$
$998$	0	0
$999$	−4.27492	−0.135252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bh.1.1		2
4.3	odd	2		546.2.a.h.1.1	✓	2
12.11	even	2		1638.2.a.y.1.2		2
28.27	even	2		3822.2.a.bm.1.2		2
52.51	odd	2		7098.2.a.bu.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.h.1.1	✓	2	4.3	odd	2
1638.2.a.y.1.2		2	12.11	even	2
3822.2.a.bm.1.2		2	28.27	even	2
4368.2.a.bh.1.1		2	1.1	even	1	trivial
7098.2.a.bu.1.2		2	52.51	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.27492 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.27492 q^{11} -1.00000 q^{13} -4.27492 q^{15} +0.274917 q^{17} +2.27492 q^{19} -1.00000 q^{21} -2.27492 q^{23} +13.2749 q^{25} +1.00000 q^{27} +8.27492 q^{29} -8.00000 q^{31} +2.27492 q^{33} +4.27492 q^{35} -4.27492 q^{37} -1.00000 q^{39} +6.54983 q^{41} +2.27492 q^{43} -4.27492 q^{45} +1.00000 q^{49} +0.274917 q^{51} +10.0000 q^{53} -9.72508 q^{55} +2.27492 q^{57} -8.00000 q^{59} -12.2749 q^{61} -1.00000 q^{63} +4.27492 q^{65} -12.5498 q^{67} -2.27492 q^{69} -12.8248 q^{73} +13.2749 q^{75} -2.27492 q^{77} -12.5498 q^{79} +1.00000 q^{81} +4.54983 q^{83} -1.17525 q^{85} +8.27492 q^{87} -14.0000 q^{89} +1.00000 q^{91} -8.00000 q^{93} -9.72508 q^{95} +15.0997 q^{97} +2.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9} - 3 q^{11} - 2 q^{13} - q^{15} - 7 q^{17} - 3 q^{19} - 2 q^{21} + 3 q^{23} + 19 q^{25} + 2 q^{27} + 9 q^{29} - 16 q^{31} - 3 q^{33} + q^{35} - q^{37} - 2 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - q^5 - 2 * q^7 + 2 * q^9 - 3 * q^11 - 2 * q^13 - q^15 - 7 * q^17 - 3 * q^19 - 2 * q^21 + 3 * q^23 + 19 * q^25 + 2 * q^27 + 9 * q^29 - 16 * q^31 - 3 * q^33 + q^35 - q^37 - 2 * q^39 - 2 * q^41 - 3 * q^43 - q^45 + 2 * q^49 - 7 * q^51 + 20 * q^53 - 27 * q^55 - 3 * q^57 - 16 * q^59 - 17 * q^61 - 2 * q^63 + q^65 - 10 * q^67 + 3 * q^69 - 3 * q^73 + 19 * q^75 + 3 * q^77 - 10 * q^79 + 2 * q^81 - 6 * q^83 - 25 * q^85 + 9 * q^87 - 28 * q^89 + 2 * q^91 - 16 * q^93 - 27 * q^95 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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