LMFDB - Embedded newform 4368.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	yes
Analytic conductor:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
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			1.2
Root			$-2.70156$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +0.701562 q^{11} +1.00000 q^{13} +2.70156 q^{15} -2.70156 q^{17} +0.701562 q^{19} -1.00000 q^{21} -4.70156 q^{23} +2.29844 q^{25} +1.00000 q^{27} +2.70156 q^{29} +0.701562 q^{33} -2.70156 q^{35} +10.7016 q^{37} +1.00000 q^{39} +3.40312 q^{41} +10.1047 q^{43} +2.70156 q^{45} +8.00000 q^{47} +1.00000 q^{49} -2.70156 q^{51} -2.00000 q^{53} +1.89531 q^{55} +0.701562 q^{57} +14.8062 q^{59} +1.29844 q^{61} -1.00000 q^{63} +2.70156 q^{65} -5.40312 q^{67} -4.70156 q^{69} +8.00000 q^{71} -1.29844 q^{73} +2.29844 q^{75} -0.701562 q^{77} -9.40312 q^{79} +1.00000 q^{81} +13.4031 q^{83} -7.29844 q^{85} +2.70156 q^{87} -8.80625 q^{89} -1.00000 q^{91} +1.89531 q^{95} -8.80625 q^{97} +0.701562 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9} - 5 q^{11} + 2 q^{13} - q^{15} + q^{17} - 5 q^{19} - 2 q^{21} - 3 q^{23} + 11 q^{25} + 2 q^{27} - q^{29} - 5 q^{33} + q^{35} + 15 q^{37} + 2 q^{39} - 6 q^{41} + q^{43} - q^{45} + 16 q^{47} + 2 q^{49} + q^{51} - 4 q^{53} + 23 q^{55} - 5 q^{57} + 4 q^{59} + 9 q^{61} - 2 q^{63} - q^{65} + 2 q^{67} - 3 q^{69} + 16 q^{71} - 9 q^{73} + 11 q^{75} + 5 q^{77} - 6 q^{79} + 2 q^{81} + 14 q^{83} - 21 q^{85} - q^{87} + 8 q^{89} - 2 q^{91} + 23 q^{95} + 8 q^{97} - 5 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - q^5 - 2 * q^7 + 2 * q^9 - 5 * q^11 + 2 * q^13 - q^15 + q^17 - 5 * q^19 - 2 * q^21 - 3 * q^23 + 11 * q^25 + 2 * q^27 - q^29 - 5 * q^33 + q^35 + 15 * q^37 + 2 * q^39 - 6 * q^41 + q^43 - q^45 + 16 * q^47 + 2 * q^49 + q^51 - 4 * q^53 + 23 * q^55 - 5 * q^57 + 4 * q^59 + 9 * q^61 - 2 * q^63 - q^65 + 2 * q^67 - 3 * q^69 + 16 * q^71 - 9 * q^73 + 11 * q^75 + 5 * q^77 - 6 * q^79 + 2 * q^81 + 14 * q^83 - 21 * q^85 - q^87 + 8 * q^89 - 2 * q^91 + 23 * q^95 + 8 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

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
$5$	2.70156	1.20818	0.604088	−	0.796918i	$-0.293538\pi$
$5$	2.70156	1.20818	0.604088	+	0.796918i	$0.293538\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.701562	0.211529	0.105764	−	0.994391i	$-0.466271\pi$
$11$	0.701562	0.211529	0.105764	+	0.994391i	$0.466271\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.70156	0.697540
$16$	0	0
$17$	−2.70156	−0.655225	−0.327613	−	0.944812i	$-0.606244\pi$
$17$	−2.70156	−0.655225	−0.327613	+	0.944812i	$0.606244\pi$
$18$	0	0
$19$	0.701562	0.160949	0.0804747	−	0.996757i	$-0.474356\pi$
$19$	0.701562	0.160949	0.0804747	+	0.996757i	$0.474356\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−4.70156	−0.980343	−0.490172	−	0.871626i	$-0.663066\pi$
$23$	−4.70156	−0.980343	−0.490172	+	0.871626i	$0.663066\pi$
$24$	0	0
$25$	2.29844	0.459688
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.70156	0.501667	0.250834	−	0.968030i	$-0.419295\pi$
$29$	2.70156	0.501667	0.250834	+	0.968030i	$0.419295\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0.701562	0.122126
$34$	0	0
$35$	−2.70156	−0.456647
$36$	0	0
$37$	10.7016	1.75933	0.879663	−	0.475598i	$-0.157768\pi$
$37$	10.7016	1.75933	0.879663	+	0.475598i	$0.157768\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	3.40312	0.531479	0.265739	−	0.964045i	$-0.414384\pi$
$41$	3.40312	0.531479	0.265739	+	0.964045i	$0.414384\pi$
$42$	0	0
$43$	10.1047	1.54095	0.770475	−	0.637470i	$-0.220019\pi$
$43$	10.1047	1.54095	0.770475	+	0.637470i	$0.220019\pi$
$44$	0	0
$45$	2.70156	0.402725
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.70156	−0.378294
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	1.89531	0.255564
$56$	0	0
$57$	0.701562	0.0929242
$58$	0	0
$59$	14.8062	1.92761	0.963805	−	0.266609i	$-0.0859033\pi$
$59$	14.8062	1.92761	0.963805	+	0.266609i	$0.0859033\pi$
$60$	0	0
$61$	1.29844	0.166248	0.0831240	−	0.996539i	$-0.473510\pi$
$61$	1.29844	0.166248	0.0831240	+	0.996539i	$0.473510\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	2.70156	0.335088
$66$	0	0
$67$	−5.40312	−0.660097	−0.330048	−	0.943964i	$-0.607065\pi$
$67$	−5.40312	−0.660097	−0.330048	+	0.943964i	$0.607065\pi$
$68$	0	0
$69$	−4.70156	−0.566002
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−1.29844	−0.151971	−0.0759853	−	0.997109i	$-0.524210\pi$
$73$	−1.29844	−0.151971	−0.0759853	+	0.997109i	$0.524210\pi$
$74$	0	0
$75$	2.29844	0.265401
$76$	0	0
$77$	−0.701562	−0.0799504
$78$	0	0
$79$	−9.40312	−1.05793	−0.528967	−	0.848642i	$-0.677421\pi$
$79$	−9.40312	−1.05793	−0.528967	+	0.848642i	$0.677421\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.4031	1.47118	0.735592	−	0.677425i	$-0.236904\pi$
$83$	13.4031	1.47118	0.735592	+	0.677425i	$0.236904\pi$
$84$	0	0
$85$	−7.29844	−0.791627
$86$	0	0
$87$	2.70156	0.289638
$88$	0	0
$89$	−8.80625	−0.933460	−0.466730	−	0.884400i	$-0.654568\pi$
$89$	−8.80625	−0.933460	−0.466730	+	0.884400i	$0.654568\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.89531	0.194455
$96$	0	0
$97$	−8.80625	−0.894139	−0.447070	−	0.894499i	$-0.647532\pi$
$97$	−8.80625	−0.894139	−0.447070	+	0.894499i	$0.647532\pi$
$98$	0	0
$99$	0.701562	0.0705096
$100$	0	0
$101$	−3.40312	−0.338624	−0.169312	−	0.985563i	$-0.554154\pi$
$101$	−3.40312	−0.338624	−0.169312	+	0.985563i	$0.554154\pi$
$102$	0	0
$103$	3.29844	0.325005	0.162502	−	0.986708i	$-0.448043\pi$
$103$	3.29844	0.325005	0.162502	+	0.986708i	$0.448043\pi$
$104$	0	0
$105$	−2.70156	−0.263645
$106$	0	0
$107$	5.40312	0.522340	0.261170	−	0.965293i	$-0.415892\pi$
$107$	5.40312	0.522340	0.261170	+	0.965293i	$0.415892\pi$
$108$	0	0
$109$	9.29844	0.890629	0.445314	−	0.895374i	$-0.353092\pi$
$109$	9.29844	0.890629	0.445314	+	0.895374i	$0.353092\pi$
$110$	0	0
$111$	10.7016	1.01575
$112$	0	0
$113$	4.80625	0.452134	0.226067	−	0.974112i	$-0.427413\pi$
$113$	4.80625	0.452134	0.226067	+	0.974112i	$0.427413\pi$
$114$	0	0
$115$	−12.7016	−1.18443
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	2.70156	0.247652
$120$	0	0
$121$	−10.5078	−0.955256
$122$	0	0
$123$	3.40312	0.306849
$124$	0	0
$125$	−7.29844	−0.652792
$126$	0	0
$127$	6.59688	0.585378	0.292689	−	0.956208i	$-0.405450\pi$
$127$	6.59688	0.585378	0.292689	+	0.956208i	$0.405450\pi$
$128$	0	0
$129$	10.1047	0.889668
$130$	0	0
$131$	7.29844	0.637667	0.318834	−	0.947811i	$-0.396709\pi$
$131$	7.29844	0.637667	0.318834	+	0.947811i	$0.396709\pi$
$132$	0	0
$133$	−0.701562	−0.0608332
$134$	0	0
$135$	2.70156	0.232513
$136$	0	0
$137$	−18.7016	−1.59778	−0.798891	−	0.601476i	$-0.794580\pi$
$137$	−18.7016	−1.59778	−0.798891	+	0.601476i	$0.794580\pi$
$138$	0	0
$139$	−6.80625	−0.577298	−0.288649	−	0.957435i	$-0.593206\pi$
$139$	−6.80625	−0.577298	−0.288649	+	0.957435i	$0.593206\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	0.701562	0.0586676
$144$	0	0
$145$	7.29844	0.606102
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	15.4031	1.26187	0.630937	−	0.775834i	$-0.282671\pi$
$149$	15.4031	1.26187	0.630937	+	0.775834i	$0.282671\pi$
$150$	0	0
$151$	4.70156	0.382608	0.191304	−	0.981531i	$-0.438728\pi$
$151$	4.70156	0.382608	0.191304	+	0.981531i	$0.438728\pi$
$152$	0	0
$153$	−2.70156	−0.218408
$154$	0	0
$155$	0	0
$156$	0	0
$157$	20.1047	1.60453	0.802264	−	0.596969i	$-0.203628\pi$
$157$	20.1047	1.60453	0.802264	+	0.596969i	$0.203628\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	4.70156	0.370535
$162$	0	0
$163$	−5.40312	−0.423205	−0.211603	−	0.977356i	$-0.567868\pi$
$163$	−5.40312	−0.423205	−0.211603	+	0.977356i	$0.567868\pi$
$164$	0	0
$165$	1.89531	0.147550
$166$	0	0
$167$	−3.29844	−0.255241	−0.127620	−	0.991823i	$-0.540734\pi$
$167$	−3.29844	−0.255241	−0.127620	+	0.991823i	$0.540734\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.701562	0.0536498
$172$	0	0
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	−2.29844	−0.173746
$176$	0	0
$177$	14.8062	1.11291
$178$	0	0
$179$	−14.8062	−1.10667	−0.553335	−	0.832958i	$-0.686645\pi$
$179$	−14.8062	−1.10667	−0.553335	+	0.832958i	$0.686645\pi$
$180$	0	0
$181$	8.80625	0.654563	0.327282	−	0.944927i	$-0.393867\pi$
$181$	8.80625	0.654563	0.327282	+	0.944927i	$0.393867\pi$
$182$	0	0
$183$	1.29844	0.0959833
$184$	0	0
$185$	28.9109	2.12557
$186$	0	0
$187$	−1.89531	−0.138599
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−12.7016	−0.919053	−0.459526	−	0.888164i	$-0.651981\pi$
$191$	−12.7016	−0.919053	−0.459526	+	0.888164i	$0.651981\pi$
$192$	0	0
$193$	11.4031	0.820815	0.410407	−	0.911902i	$-0.365386\pi$
$193$	11.4031	0.820815	0.410407	+	0.911902i	$0.365386\pi$
$194$	0	0
$195$	2.70156	0.193463
$196$	0	0
$197$	−3.40312	−0.242463	−0.121231	−	0.992624i	$-0.538684\pi$
$197$	−3.40312	−0.242463	−0.121231	+	0.992624i	$0.538684\pi$
$198$	0	0
$199$	−22.1047	−1.56696	−0.783480	−	0.621417i	$-0.786557\pi$
$199$	−22.1047	−1.56696	−0.783480	+	0.621417i	$0.786557\pi$
$200$	0	0
$201$	−5.40312	−0.381107
$202$	0	0
$203$	−2.70156	−0.189612
$204$	0	0
$205$	9.19375	0.642119
$206$	0	0
$207$	−4.70156	−0.326781
$208$	0	0
$209$	0.492189	0.0340455
$210$	0	0
$211$	24.7016	1.70053	0.850263	−	0.526358i	$-0.176443\pi$
$211$	24.7016	1.70053	0.850263	+	0.526358i	$0.176443\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	27.2984	1.86174
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.29844	−0.0877403
$220$	0	0
$221$	−2.70156	−0.181727
$222$	0	0
$223$	−9.40312	−0.629680	−0.314840	−	0.949145i	$-0.601951\pi$
$223$	−9.40312	−0.629680	−0.314840	+	0.949145i	$0.601951\pi$
$224$	0	0
$225$	2.29844	0.153229
$226$	0	0
$227$	−21.4031	−1.42058	−0.710288	−	0.703912i	$-0.751435\pi$
$227$	−21.4031	−1.42058	−0.710288	+	0.703912i	$0.751435\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	−0.701562	−0.0461594
$232$	0	0
$233$	−18.2094	−1.19294	−0.596468	−	0.802637i	$-0.703430\pi$
$233$	−18.2094	−1.19294	−0.596468	+	0.802637i	$0.703430\pi$
$234$	0	0
$235$	21.6125	1.40984
$236$	0	0
$237$	−9.40312	−0.610799
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−24.8062	−1.59791	−0.798955	−	0.601390i	$-0.794614\pi$
$241$	−24.8062	−1.59791	−0.798955	+	0.601390i	$0.794614\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.70156	0.172596
$246$	0	0
$247$	0.701562	0.0446393
$248$	0	0
$249$	13.4031	0.849388
$250$	0	0
$251$	−3.50781	−0.221411	−0.110706	−	0.993853i	$-0.535311\pi$
$251$	−3.50781	−0.221411	−0.110706	+	0.993853i	$0.535311\pi$
$252$	0	0
$253$	−3.29844	−0.207371
$254$	0	0
$255$	−7.29844	−0.457046
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−10.7016	−0.664963
$260$	0	0
$261$	2.70156	0.167222
$262$	0	0
$263$	26.8062	1.65294	0.826472	−	0.562978i	$-0.190344\pi$
$263$	26.8062	1.65294	0.826472	+	0.562978i	$0.190344\pi$
$264$	0	0
$265$	−5.40312	−0.331911
$266$	0	0
$267$	−8.80625	−0.538934
$268$	0	0
$269$	−4.80625	−0.293042	−0.146521	−	0.989208i	$-0.546808\pi$
$269$	−4.80625	−0.293042	−0.146521	+	0.989208i	$0.546808\pi$
$270$	0	0
$271$	−12.2094	−0.741667	−0.370833	−	0.928699i	$-0.620928\pi$
$271$	−12.2094	−0.741667	−0.370833	+	0.928699i	$0.620928\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	1.61250	0.0972372
$276$	0	0
$277$	27.6125	1.65907	0.829537	−	0.558452i	$-0.188604\pi$
$277$	27.6125	1.65907	0.829537	+	0.558452i	$0.188604\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.8062	0.763957	0.381978	−	0.924171i	$-0.375243\pi$
$281$	12.8062	0.763957	0.381978	+	0.924171i	$0.375243\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	1.89531	0.112269
$286$	0	0
$287$	−3.40312	−0.200880
$288$	0	0
$289$	−9.70156	−0.570680
$290$	0	0
$291$	−8.80625	−0.516231
$292$	0	0
$293$	−12.8062	−0.748149	−0.374075	−	0.927399i	$-0.622040\pi$
$293$	−12.8062	−0.748149	−0.374075	+	0.927399i	$0.622040\pi$
$294$	0	0
$295$	40.0000	2.32889
$296$	0	0
$297$	0.701562	0.0407088
$298$	0	0
$299$	−4.70156	−0.271898
$300$	0	0
$301$	−10.1047	−0.582424
$302$	0	0
$303$	−3.40312	−0.195504
$304$	0	0
$305$	3.50781	0.200857
$306$	0	0
$307$	6.80625	0.388453	0.194227	−	0.980957i	$-0.437780\pi$
$307$	6.80625	0.388453	0.194227	+	0.980957i	$0.437780\pi$
$308$	0	0
$309$	3.29844	0.187642
$310$	0	0
$311$	−14.5969	−0.827713	−0.413856	−	0.910342i	$-0.635818\pi$
$311$	−14.5969	−0.827713	−0.413856	+	0.910342i	$0.635818\pi$
$312$	0	0
$313$	22.2094	1.25535	0.627674	−	0.778476i	$-0.284007\pi$
$313$	22.2094	1.25535	0.627674	+	0.778476i	$0.284007\pi$
$314$	0	0
$315$	−2.70156	−0.152216
$316$	0	0
$317$	7.40312	0.415801	0.207900	−	0.978150i	$-0.433337\pi$
$317$	7.40312	0.415801	0.207900	+	0.978150i	$0.433337\pi$
$318$	0	0
$319$	1.89531	0.106117
$320$	0	0
$321$	5.40312	0.301573
$322$	0	0
$323$	−1.89531	−0.105458
$324$	0	0
$325$	2.29844	0.127494
$326$	0	0
$327$	9.29844	0.514205
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−32.2094	−1.77039	−0.885194	−	0.465223i	$-0.845974\pi$
$331$	−32.2094	−1.77039	−0.885194	+	0.465223i	$0.845974\pi$
$332$	0	0
$333$	10.7016	0.586442
$334$	0	0
$335$	−14.5969	−0.797513
$336$	0	0
$337$	−29.5078	−1.60739	−0.803696	−	0.595040i	$-0.797136\pi$
$337$	−29.5078	−1.60739	−0.803696	+	0.595040i	$0.797136\pi$
$338$	0	0
$339$	4.80625	0.261040
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−12.7016	−0.683829
$346$	0	0
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	20.8062	1.10740	0.553702	−	0.832715i	$-0.313215\pi$
$353$	20.8062	1.10740	0.553702	+	0.832715i	$0.313215\pi$
$354$	0	0
$355$	21.6125	1.14707
$356$	0	0
$357$	2.70156	0.142982
$358$	0	0
$359$	−26.8062	−1.41478	−0.707390	−	0.706824i	$-0.750127\pi$
$359$	−26.8062	−1.41478	−0.707390	+	0.706824i	$0.750127\pi$
$360$	0	0
$361$	−18.5078	−0.974095
$362$	0	0
$363$	−10.5078	−0.551517
$364$	0	0
$365$	−3.50781	−0.183607
$366$	0	0
$367$	29.6125	1.54576	0.772880	−	0.634552i	$-0.218816\pi$
$367$	29.6125	1.54576	0.772880	+	0.634552i	$0.218816\pi$
$368$	0	0
$369$	3.40312	0.177160
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	−19.4031	−1.00466	−0.502328	−	0.864677i	$-0.667523\pi$
$373$	−19.4031	−1.00466	−0.502328	+	0.864677i	$0.667523\pi$
$374$	0	0
$375$	−7.29844	−0.376890
$376$	0	0
$377$	2.70156	0.139138
$378$	0	0
$379$	17.6125	0.904693	0.452347	−	0.891842i	$-0.350587\pi$
$379$	17.6125	0.904693	0.452347	+	0.891842i	$0.350587\pi$
$380$	0	0
$381$	6.59688	0.337968
$382$	0	0
$383$	−16.9109	−0.864108	−0.432054	−	0.901848i	$-0.642211\pi$
$383$	−16.9109	−0.864108	−0.432054	+	0.901848i	$0.642211\pi$
$384$	0	0
$385$	−1.89531	−0.0965941
$386$	0	0
$387$	10.1047	0.513650
$388$	0	0
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	12.7016	0.642346
$392$	0	0
$393$	7.29844	0.368157
$394$	0	0
$395$	−25.4031	−1.27817
$396$	0	0
$397$	0.806248	0.0404645	0.0202322	−	0.999795i	$-0.493559\pi$
$397$	0.806248	0.0404645	0.0202322	+	0.999795i	$0.493559\pi$
$398$	0	0
$399$	−0.701562	−0.0351220
$400$	0	0
$401$	4.80625	0.240013	0.120006	−	0.992773i	$-0.461709\pi$
$401$	4.80625	0.240013	0.120006	+	0.992773i	$0.461709\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.70156	0.134242
$406$	0	0
$407$	7.50781	0.372148
$408$	0	0
$409$	5.29844	0.261991	0.130995	−	0.991383i	$-0.458183\pi$
$409$	5.29844	0.261991	0.130995	+	0.991383i	$0.458183\pi$
$410$	0	0
$411$	−18.7016	−0.922480
$412$	0	0
$413$	−14.8062	−0.728568
$414$	0	0
$415$	36.2094	1.77745
$416$	0	0
$417$	−6.80625	−0.333303
$418$	0	0
$419$	−34.1047	−1.66612	−0.833061	−	0.553180i	$-0.813414\pi$
$419$	−34.1047	−1.66612	−0.833061	+	0.553180i	$0.813414\pi$
$420$	0	0
$421$	19.6125	0.955855	0.477927	−	0.878399i	$-0.341388\pi$
$421$	19.6125	0.955855	0.477927	+	0.878399i	$0.341388\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	−6.20937	−0.301199
$426$	0	0
$427$	−1.29844	−0.0628358
$428$	0	0
$429$	0.701562	0.0338717
$430$	0	0
$431$	12.2094	0.588105	0.294052	−	0.955789i	$-0.404996\pi$
$431$	12.2094	0.588105	0.294052	+	0.955789i	$0.404996\pi$
$432$	0	0
$433$	14.2094	0.682859	0.341429	−	0.939907i	$-0.389089\pi$
$433$	14.2094	0.682859	0.341429	+	0.939907i	$0.389089\pi$
$434$	0	0
$435$	7.29844	0.349933
$436$	0	0
$437$	−3.29844	−0.157786
$438$	0	0
$439$	0.492189	0.0234909	0.0117455	−	0.999931i	$-0.496261\pi$
$439$	0.492189	0.0234909	0.0117455	+	0.999931i	$0.496261\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−0.209373	−0.00994760	−0.00497380	−	0.999988i	$-0.501583\pi$
$443$	−0.209373	−0.00994760	−0.00497380	+	0.999988i	$0.501583\pi$
$444$	0	0
$445$	−23.7906	−1.12778
$446$	0	0
$447$	15.4031	0.728543
$448$	0	0
$449$	−7.89531	−0.372603	−0.186301	−	0.982493i	$-0.559650\pi$
$449$	−7.89531	−0.372603	−0.186301	+	0.982493i	$0.559650\pi$
$450$	0	0
$451$	2.38750	0.112423
$452$	0	0
$453$	4.70156	0.220899
$454$	0	0
$455$	−2.70156	−0.126651
$456$	0	0
$457$	0.596876	0.0279207	0.0139603	−	0.999903i	$-0.495556\pi$
$457$	0.596876	0.0279207	0.0139603	+	0.999903i	$0.495556\pi$
$458$	0	0
$459$	−2.70156	−0.126098
$460$	0	0
$461$	−20.3141	−0.946120	−0.473060	−	0.881030i	$-0.656851\pi$
$461$	−20.3141	−0.946120	−0.473060	+	0.881030i	$0.656851\pi$
$462$	0	0
$463$	34.3141	1.59471	0.797355	−	0.603511i	$-0.206232\pi$
$463$	34.3141	1.59471	0.797355	+	0.603511i	$0.206232\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.49219	0.207874	0.103937	−	0.994584i	$-0.466856\pi$
$467$	4.49219	0.207874	0.103937	+	0.994584i	$0.466856\pi$
$468$	0	0
$469$	5.40312	0.249493
$470$	0	0
$471$	20.1047	0.926375
$472$	0	0
$473$	7.08907	0.325956
$474$	0	0
$475$	1.61250	0.0739864
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−7.50781	−0.343041	−0.171520	−	0.985181i	$-0.554868\pi$
$479$	−7.50781	−0.343041	−0.171520	+	0.985181i	$0.554868\pi$
$480$	0	0
$481$	10.7016	0.487949
$482$	0	0
$483$	4.70156	0.213928
$484$	0	0
$485$	−23.7906	−1.08028
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	−5.40312	−0.244338
$490$	0	0
$491$	−26.5969	−1.20030	−0.600150	−	0.799887i	$-0.704892\pi$
$491$	−26.5969	−1.20030	−0.600150	+	0.799887i	$0.704892\pi$
$492$	0	0
$493$	−7.29844	−0.328705
$494$	0	0
$495$	1.89531	0.0851880
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	1.19375	0.0534397	0.0267198	−	0.999643i	$-0.491494\pi$
$499$	1.19375	0.0534397	0.0267198	+	0.999643i	$0.491494\pi$
$500$	0	0
$501$	−3.29844	−0.147363
$502$	0	0
$503$	33.4031	1.48937	0.744686	−	0.667415i	$-0.232599\pi$
$503$	33.4031	1.48937	0.744686	+	0.667415i	$0.232599\pi$
$504$	0	0
$505$	−9.19375	−0.409117
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	38.9109	1.72470	0.862348	−	0.506315i	$-0.168993\pi$
$509$	38.9109	1.72470	0.862348	+	0.506315i	$0.168993\pi$
$510$	0	0
$511$	1.29844	0.0574395
$512$	0	0
$513$	0.701562	0.0309747
$514$	0	0
$515$	8.91093	0.392663
$516$	0	0
$517$	5.61250	0.246837
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−1.29844	−0.0568856	−0.0284428	−	0.999595i	$-0.509055\pi$
$521$	−1.29844	−0.0568856	−0.0284428	+	0.999595i	$0.509055\pi$
$522$	0	0
$523$	33.6125	1.46977	0.734886	−	0.678191i	$-0.237236\pi$
$523$	33.6125	1.46977	0.734886	+	0.678191i	$0.237236\pi$
$524$	0	0
$525$	−2.29844	−0.100312
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−0.895314	−0.0389267
$530$	0	0
$531$	14.8062	0.642536
$532$	0	0
$533$	3.40312	0.147406
$534$	0	0
$535$	14.5969	0.631078
$536$	0	0
$537$	−14.8062	−0.638937
$538$	0	0
$539$	0.701562	0.0302184
$540$	0	0
$541$	−6.70156	−0.288123	−0.144061	−	0.989569i	$-0.546016\pi$
$541$	−6.70156	−0.288123	−0.144061	+	0.989569i	$0.546016\pi$
$542$	0	0
$543$	8.80625	0.377912
$544$	0	0
$545$	25.1203	1.07604
$546$	0	0
$547$	−9.61250	−0.411001	−0.205500	−	0.978657i	$-0.565882\pi$
$547$	−9.61250	−0.411001	−0.205500	+	0.978657i	$0.565882\pi$
$548$	0	0
$549$	1.29844	0.0554160
$550$	0	0
$551$	1.89531	0.0807431
$552$	0	0
$553$	9.40312	0.399862
$554$	0	0
$555$	28.9109	1.22720
$556$	0	0
$557$	−24.5969	−1.04220	−0.521102	−	0.853495i	$-0.674479\pi$
$557$	−24.5969	−1.04220	−0.521102	+	0.853495i	$0.674479\pi$
$558$	0	0
$559$	10.1047	0.427383
$560$	0	0
$561$	−1.89531	−0.0800202
$562$	0	0
$563$	−8.70156	−0.366727	−0.183364	−	0.983045i	$-0.558699\pi$
$563$	−8.70156	−0.366727	−0.183364	+	0.983045i	$0.558699\pi$
$564$	0	0
$565$	12.9844	0.546257
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	1.19375	0.0499569	0.0249785	−	0.999688i	$-0.492048\pi$
$571$	1.19375	0.0499569	0.0249785	+	0.999688i	$0.492048\pi$
$572$	0	0
$573$	−12.7016	−0.530615
$574$	0	0
$575$	−10.8062	−0.450652
$576$	0	0
$577$	−8.80625	−0.366609	−0.183304	−	0.983056i	$-0.558679\pi$
$577$	−8.80625	−0.366609	−0.183304	+	0.983056i	$0.558679\pi$
$578$	0	0
$579$	11.4031	0.473898
$580$	0	0
$581$	−13.4031	−0.556055
$582$	0	0
$583$	−1.40312	−0.0581115
$584$	0	0
$585$	2.70156	0.111696
$586$	0	0
$587$	−35.0156	−1.44525	−0.722625	−	0.691241i	$-0.757064\pi$
$587$	−35.0156	−1.44525	−0.722625	+	0.691241i	$0.757064\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−3.40312	−0.139986
$592$	0	0
$593$	−16.8062	−0.690150	−0.345075	−	0.938575i	$-0.612147\pi$
$593$	−16.8062	−0.690150	−0.345075	+	0.938575i	$0.612147\pi$
$594$	0	0
$595$	7.29844	0.299207
$596$	0	0
$597$	−22.1047	−0.904685
$598$	0	0
$599$	−11.2984	−0.461642	−0.230821	−	0.972996i	$-0.574141\pi$
$599$	−11.2984	−0.461642	−0.230821	+	0.972996i	$0.574141\pi$
$600$	0	0
$601$	−15.4031	−0.628307	−0.314153	−	0.949372i	$-0.601721\pi$
$601$	−15.4031	−0.628307	−0.314153	+	0.949372i	$0.601721\pi$
$602$	0	0
$603$	−5.40312	−0.220032
$604$	0	0
$605$	−28.3875	−1.15412
$606$	0	0
$607$	−7.50781	−0.304733	−0.152366	−	0.988324i	$-0.548689\pi$
$607$	−7.50781	−0.304733	−0.152366	+	0.988324i	$0.548689\pi$
$608$	0	0
$609$	−2.70156	−0.109473
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	38.9109	1.57160	0.785799	−	0.618482i	$-0.212252\pi$
$613$	38.9109	1.57160	0.785799	+	0.618482i	$0.212252\pi$
$614$	0	0
$615$	9.19375	0.370728
$616$	0	0
$617$	−30.9109	−1.24443	−0.622214	−	0.782847i	$-0.713767\pi$
$617$	−30.9109	−1.24443	−0.622214	+	0.782847i	$0.713767\pi$
$618$	0	0
$619$	−7.29844	−0.293349	−0.146674	−	0.989185i	$-0.546857\pi$
$619$	−7.29844	−0.293349	−0.146674	+	0.989185i	$0.546857\pi$
$620$	0	0
$621$	−4.70156	−0.188667
$622$	0	0
$623$	8.80625	0.352815
$624$	0	0
$625$	−31.2094	−1.24837
$626$	0	0
$627$	0.492189	0.0196562
$628$	0	0
$629$	−28.9109	−1.15275
$630$	0	0
$631$	7.50781	0.298881	0.149441	−	0.988771i	$-0.452253\pi$
$631$	7.50781	0.298881	0.149441	+	0.988771i	$0.452253\pi$
$632$	0	0
$633$	24.7016	0.981799
$634$	0	0
$635$	17.8219	0.707239
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−48.8062	−1.92773	−0.963865	−	0.266390i	$-0.914169\pi$
$641$	−48.8062	−1.92773	−0.963865	+	0.266390i	$0.914169\pi$
$642$	0	0
$643$	−46.3141	−1.82645	−0.913224	−	0.407458i	$-0.866415\pi$
$643$	−46.3141	−1.82645	−0.913224	+	0.407458i	$0.866415\pi$
$644$	0	0
$645$	27.2984	1.07487
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	10.3875	0.407745
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.50781	−0.372069	−0.186035	−	0.982543i	$-0.559564\pi$
$653$	−9.50781	−0.372069	−0.186035	+	0.982543i	$0.559564\pi$
$654$	0	0
$655$	19.7172	0.770414
$656$	0	0
$657$	−1.29844	−0.0506569
$658$	0	0
$659$	35.0156	1.36401	0.682007	−	0.731345i	$-0.261107\pi$
$659$	35.0156	1.36401	0.682007	+	0.731345i	$0.261107\pi$
$660$	0	0
$661$	−50.4187	−1.96106	−0.980531	−	0.196365i	$-0.937086\pi$
$661$	−50.4187	−1.96106	−0.980531	+	0.196365i	$0.937086\pi$
$662$	0	0
$663$	−2.70156	−0.104920
$664$	0	0
$665$	−1.89531	−0.0734971
$666$	0	0
$667$	−12.7016	−0.491806
$668$	0	0
$669$	−9.40312	−0.363546
$670$	0	0
$671$	0.910935	0.0351662
$672$	0	0
$673$	42.9109	1.65409	0.827047	−	0.562132i	$-0.190019\pi$
$673$	42.9109	1.65409	0.827047	+	0.562132i	$0.190019\pi$
$674$	0	0
$675$	2.29844	0.0884669
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	8.80625	0.337953
$680$	0	0
$681$	−21.4031	−0.820170
$682$	0	0
$683$	−11.5078	−0.440334	−0.220167	−	0.975462i	$-0.570660\pi$
$683$	−11.5078	−0.440334	−0.220167	+	0.975462i	$0.570660\pi$
$684$	0	0
$685$	−50.5234	−1.93040
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	−49.6125	−1.88735	−0.943674	−	0.330876i	$-0.892656\pi$
$691$	−49.6125	−1.88735	−0.943674	+	0.330876i	$0.892656\pi$
$692$	0	0
$693$	−0.701562	−0.0266501
$694$	0	0
$695$	−18.3875	−0.697478
$696$	0	0
$697$	−9.19375	−0.348238
$698$	0	0
$699$	−18.2094	−0.688742
$700$	0	0
$701$	3.19375	0.120626	0.0603132	−	0.998180i	$-0.480790\pi$
$701$	3.19375	0.120626	0.0603132	+	0.998180i	$0.480790\pi$
$702$	0	0
$703$	7.50781	0.283162
$704$	0	0
$705$	21.6125	0.813974
$706$	0	0
$707$	3.40312	0.127988
$708$	0	0
$709$	51.6125	1.93835	0.969174	−	0.246377i	$-0.0792402\pi$
$709$	51.6125	1.93835	0.969174	+	0.246377i	$0.0792402\pi$
$710$	0	0
$711$	−9.40312	−0.352645
$712$	0	0
$713$	0	0
$714$	0	0
$715$	1.89531	0.0708807
$716$	0	0
$717$	−16.0000	−0.597531
$718$	0	0
$719$	31.0156	1.15669	0.578344	−	0.815793i	$-0.303699\pi$
$719$	31.0156	1.15669	0.578344	+	0.815793i	$0.303699\pi$
$720$	0	0
$721$	−3.29844	−0.122840
$722$	0	0
$723$	−24.8062	−0.922554
$724$	0	0
$725$	6.20937	0.230610
$726$	0	0
$727$	22.1047	0.819817	0.409909	−	0.912127i	$-0.365561\pi$
$727$	22.1047	0.819817	0.409909	+	0.912127i	$0.365561\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−27.2984	−1.00967
$732$	0	0
$733$	20.5969	0.760763	0.380381	−	0.924830i	$-0.375793\pi$
$733$	20.5969	0.760763	0.380381	+	0.924830i	$0.375793\pi$
$734$	0	0
$735$	2.70156	0.0996486
$736$	0	0
$737$	−3.79063	−0.139630
$738$	0	0
$739$	−40.2094	−1.47913	−0.739563	−	0.673088i	$-0.764968\pi$
$739$	−40.2094	−1.47913	−0.739563	+	0.673088i	$0.764968\pi$
$740$	0	0
$741$	0.701562	0.0257725
$742$	0	0
$743$	−23.0156	−0.844361	−0.422181	−	0.906512i	$-0.638735\pi$
$743$	−23.0156	−0.844361	−0.422181	+	0.906512i	$0.638735\pi$
$744$	0	0
$745$	41.6125	1.52456
$746$	0	0
$747$	13.4031	0.490395
$748$	0	0
$749$	−5.40312	−0.197426
$750$	0	0
$751$	−34.8062	−1.27010	−0.635049	−	0.772472i	$-0.719020\pi$
$751$	−34.8062	−1.27010	−0.635049	+	0.772472i	$0.719020\pi$
$752$	0	0
$753$	−3.50781	−0.127832
$754$	0	0
$755$	12.7016	0.462257
$756$	0	0
$757$	30.4187	1.10559	0.552794	−	0.833318i	$-0.313562\pi$
$757$	30.4187	1.10559	0.552794	+	0.833318i	$0.313562\pi$
$758$	0	0
$759$	−3.29844	−0.119726
$760$	0	0
$761$	32.5969	1.18164	0.590818	−	0.806805i	$-0.298805\pi$
$761$	32.5969	1.18164	0.590818	+	0.806805i	$0.298805\pi$
$762$	0	0
$763$	−9.29844	−0.336626
$764$	0	0
$765$	−7.29844	−0.263876
$766$	0	0
$767$	14.8062	0.534623
$768$	0	0
$769$	50.9109	1.83590	0.917948	−	0.396702i	$-0.129845\pi$
$769$	50.9109	1.83590	0.917948	+	0.396702i	$0.129845\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	0	0
$773$	−28.3141	−1.01839	−0.509193	−	0.860652i	$-0.670056\pi$
$773$	−28.3141	−1.01839	−0.509193	+	0.860652i	$0.670056\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−10.7016	−0.383916
$778$	0	0
$779$	2.38750	0.0855412
$780$	0	0
$781$	5.61250	0.200831
$782$	0	0
$783$	2.70156	0.0965460
$784$	0	0
$785$	54.3141	1.93855
$786$	0	0
$787$	−20.9109	−0.745394	−0.372697	−	0.927953i	$-0.621567\pi$
$787$	−20.9109	−0.745394	−0.372697	+	0.927953i	$0.621567\pi$
$788$	0	0
$789$	26.8062	0.954328
$790$	0	0
$791$	−4.80625	−0.170891
$792$	0	0
$793$	1.29844	0.0461089
$794$	0	0
$795$	−5.40312	−0.191629
$796$	0	0
$797$	−40.5969	−1.43802	−0.719008	−	0.695002i	$-0.755403\pi$
$797$	−40.5969	−1.43802	−0.719008	+	0.695002i	$0.755403\pi$
$798$	0	0
$799$	−21.6125	−0.764595
$800$	0	0
$801$	−8.80625	−0.311153
$802$	0	0
$803$	−0.910935	−0.0321462
$804$	0	0
$805$	12.7016	0.447671
$806$	0	0
$807$	−4.80625	−0.169188
$808$	0	0
$809$	−11.6125	−0.408274	−0.204137	−	0.978942i	$-0.565439\pi$
$809$	−11.6125	−0.408274	−0.204137	+	0.978942i	$0.565439\pi$
$810$	0	0
$811$	21.8953	0.768848	0.384424	−	0.923157i	$-0.374400\pi$
$811$	21.8953	0.768848	0.384424	+	0.923157i	$0.374400\pi$
$812$	0	0
$813$	−12.2094	−0.428201
$814$	0	0
$815$	−14.5969	−0.511306
$816$	0	0
$817$	7.08907	0.248015
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	28.5969	0.998038	0.499019	−	0.866591i	$-0.333694\pi$
$821$	28.5969	0.998038	0.499019	+	0.866591i	$0.333694\pi$
$822$	0	0
$823$	−5.19375	−0.181043	−0.0905214	−	0.995895i	$-0.528853\pi$
$823$	−5.19375	−0.181043	−0.0905214	+	0.995895i	$0.528853\pi$
$824$	0	0
$825$	1.61250	0.0561399
$826$	0	0
$827$	−52.9109	−1.83989	−0.919947	−	0.392043i	$-0.871768\pi$
$827$	−52.9109	−1.83989	−0.919947	+	0.392043i	$0.871768\pi$
$828$	0	0
$829$	−36.3141	−1.26124	−0.630620	−	0.776092i	$-0.717199\pi$
$829$	−36.3141	−1.26124	−0.630620	+	0.776092i	$0.717199\pi$
$830$	0	0
$831$	27.6125	0.957867
$832$	0	0
$833$	−2.70156	−0.0936036
$834$	0	0
$835$	−8.91093	−0.308376
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34.8062	−1.20165	−0.600823	−	0.799382i	$-0.705160\pi$
$839$	−34.8062	−1.20165	−0.600823	+	0.799382i	$0.705160\pi$
$840$	0	0
$841$	−21.7016	−0.748330
$842$	0	0
$843$	12.8062	0.441071
$844$	0	0
$845$	2.70156	0.0929366
$846$	0	0
$847$	10.5078	0.361053
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−50.3141	−1.72474
$852$	0	0
$853$	−22.2094	−0.760434	−0.380217	−	0.924897i	$-0.624151\pi$
$853$	−22.2094	−0.760434	−0.380217	+	0.924897i	$0.624151\pi$
$854$	0	0
$855$	1.89531	0.0648184
$856$	0	0
$857$	−16.8062	−0.574091	−0.287045	−	0.957917i	$-0.592673\pi$
$857$	−16.8062	−0.574091	−0.287045	+	0.957917i	$0.592673\pi$
$858$	0	0
$859$	−38.8062	−1.32405	−0.662026	−	0.749481i	$-0.730303\pi$
$859$	−38.8062	−1.32405	−0.662026	+	0.749481i	$0.730303\pi$
$860$	0	0
$861$	−3.40312	−0.115978
$862$	0	0
$863$	22.5969	0.769207	0.384603	−	0.923082i	$-0.374338\pi$
$863$	22.5969	0.769207	0.384603	+	0.923082i	$0.374338\pi$
$864$	0	0
$865$	−48.6281	−1.65341
$866$	0	0
$867$	−9.70156	−0.329482
$868$	0	0
$869$	−6.59688	−0.223784
$870$	0	0
$871$	−5.40312	−0.183078
$872$	0	0
$873$	−8.80625	−0.298046
$874$	0	0
$875$	7.29844	0.246732
$876$	0	0
$877$	0.387503	0.0130850	0.00654252	−	0.999979i	$-0.497917\pi$
$877$	0.387503	0.0130850	0.00654252	+	0.999979i	$0.497917\pi$
$878$	0	0
$879$	−12.8062	−0.431944
$880$	0	0
$881$	−8.31406	−0.280108	−0.140054	−	0.990144i	$-0.544728\pi$
$881$	−8.31406	−0.280108	−0.140054	+	0.990144i	$0.544728\pi$
$882$	0	0
$883$	−13.8953	−0.467615	−0.233807	−	0.972283i	$-0.575118\pi$
$883$	−13.8953	−0.467615	−0.233807	+	0.972283i	$0.575118\pi$
$884$	0	0
$885$	40.0000	1.34459
$886$	0	0
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	−6.59688	−0.221252
$890$	0	0
$891$	0.701562	0.0235032
$892$	0	0
$893$	5.61250	0.187815
$894$	0	0
$895$	−40.0000	−1.33705
$896$	0	0
$897$	−4.70156	−0.156981
$898$	0	0
$899$	0	0
$900$	0	0
$901$	5.40312	0.180004
$902$	0	0
$903$	−10.1047	−0.336263
$904$	0	0
$905$	23.7906	0.790827
$906$	0	0
$907$	25.6125	0.850449	0.425225	−	0.905088i	$-0.360195\pi$
$907$	25.6125	0.850449	0.425225	+	0.905088i	$0.360195\pi$
$908$	0	0
$909$	−3.40312	−0.112875
$910$	0	0
$911$	−40.9109	−1.35544	−0.677720	−	0.735320i	$-0.737032\pi$
$911$	−40.9109	−1.35544	−0.677720	+	0.735320i	$0.737032\pi$
$912$	0	0
$913$	9.40312	0.311198
$914$	0	0
$915$	3.50781	0.115965
$916$	0	0
$917$	−7.29844	−0.241016
$918$	0	0
$919$	14.5969	0.481507	0.240753	−	0.970586i	$-0.422606\pi$
$919$	14.5969	0.481507	0.240753	+	0.970586i	$0.422606\pi$
$920$	0	0
$921$	6.80625	0.224274
$922$	0	0
$923$	8.00000	0.263323
$924$	0	0
$925$	24.5969	0.808740
$926$	0	0
$927$	3.29844	0.108335
$928$	0	0
$929$	33.0156	1.08321	0.541604	−	0.840634i	$-0.317817\pi$
$929$	33.0156	1.08321	0.541604	+	0.840634i	$0.317817\pi$
$930$	0	0
$931$	0.701562	0.0229928
$932$	0	0
$933$	−14.5969	−0.477880
$934$	0	0
$935$	−5.12031	−0.167452
$936$	0	0
$937$	28.8062	0.941059	0.470530	−	0.882384i	$-0.344063\pi$
$937$	28.8062	0.941059	0.470530	+	0.882384i	$0.344063\pi$
$938$	0	0
$939$	22.2094	0.724775
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	0	0
$943$	−16.0000	−0.521032
$944$	0	0
$945$	−2.70156	−0.0878818
$946$	0	0
$947$	−4.49219	−0.145977	−0.0729883	−	0.997333i	$-0.523254\pi$
$947$	−4.49219	−0.145977	−0.0729883	+	0.997333i	$0.523254\pi$
$948$	0	0
$949$	−1.29844	−0.0421491
$950$	0	0
$951$	7.40312	0.240063
$952$	0	0
$953$	−39.8219	−1.28996	−0.644978	−	0.764201i	$-0.723134\pi$
$953$	−39.8219	−1.28996	−0.644978	+	0.764201i	$0.723134\pi$
$954$	0	0
$955$	−34.3141	−1.11038
$956$	0	0
$957$	1.89531	0.0612668
$958$	0	0
$959$	18.7016	0.603905
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	5.40312	0.174113
$964$	0	0
$965$	30.8062	0.991688
$966$	0	0
$967$	51.7172	1.66311	0.831556	−	0.555441i	$-0.187450\pi$
$967$	51.7172	1.66311	0.831556	+	0.555441i	$0.187450\pi$
$968$	0	0
$969$	−1.89531	−0.0608862
$970$	0	0
$971$	−54.8062	−1.75882	−0.879408	−	0.476069i	$-0.842061\pi$
$971$	−54.8062	−1.75882	−0.879408	+	0.476069i	$0.842061\pi$
$972$	0	0
$973$	6.80625	0.218198
$974$	0	0
$975$	2.29844	0.0736089
$976$	0	0
$977$	−41.7172	−1.33465	−0.667325	−	0.744766i	$-0.732561\pi$
$977$	−41.7172	−1.33465	−0.667325	+	0.744766i	$0.732561\pi$
$978$	0	0
$979$	−6.17813	−0.197454
$980$	0	0
$981$	9.29844	0.296876
$982$	0	0
$983$	38.1047	1.21535	0.607675	−	0.794186i	$-0.292102\pi$
$983$	38.1047	1.21535	0.607675	+	0.794186i	$0.292102\pi$
$984$	0	0
$985$	−9.19375	−0.292937
$986$	0	0
$987$	−8.00000	−0.254643
$988$	0	0
$989$	−47.5078	−1.51066
$990$	0	0
$991$	38.5969	1.22607	0.613035	−	0.790056i	$-0.289948\pi$
$991$	38.5969	1.22607	0.613035	+	0.790056i	$0.289948\pi$
$992$	0	0
$993$	−32.2094	−1.02213
$994$	0	0
$995$	−59.7172	−1.89316
$996$	0	0
$997$	−12.8062	−0.405578	−0.202789	−	0.979222i	$-0.565001\pi$
$997$	−12.8062	−0.405578	−0.202789	+	0.979222i	$0.565001\pi$
$998$	0	0
$999$	10.7016	0.338582

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bg.1.2		2
4.3	odd	2		546.2.a.i.1.2	✓	2
12.11	even	2		1638.2.a.w.1.1		2
28.27	even	2		3822.2.a.bt.1.1		2
52.51	odd	2		7098.2.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.i.1.2	✓	2	4.3	odd	2
1638.2.a.w.1.1		2	12.11	even	2
3822.2.a.bt.1.1		2	28.27	even	2
4368.2.a.bg.1.2		2	1.1	even	1	trivial
7098.2.a.bh.1.1		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +0.701562 q^{11} +1.00000 q^{13} +2.70156 q^{15} -2.70156 q^{17} +0.701562 q^{19} -1.00000 q^{21} -4.70156 q^{23} +2.29844 q^{25} +1.00000 q^{27} +2.70156 q^{29} +0.701562 q^{33} -2.70156 q^{35} +10.7016 q^{37} +1.00000 q^{39} +3.40312 q^{41} +10.1047 q^{43} +2.70156 q^{45} +8.00000 q^{47} +1.00000 q^{49} -2.70156 q^{51} -2.00000 q^{53} +1.89531 q^{55} +0.701562 q^{57} +14.8062 q^{59} +1.29844 q^{61} -1.00000 q^{63} +2.70156 q^{65} -5.40312 q^{67} -4.70156 q^{69} +8.00000 q^{71} -1.29844 q^{73} +2.29844 q^{75} -0.701562 q^{77} -9.40312 q^{79} +1.00000 q^{81} +13.4031 q^{83} -7.29844 q^{85} +2.70156 q^{87} -8.80625 q^{89} -1.00000 q^{91} +1.89531 q^{95} -8.80625 q^{97} +0.701562 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9} - 5 q^{11} + 2 q^{13} - q^{15} + q^{17} - 5 q^{19} - 2 q^{21} - 3 q^{23} + 11 q^{25} + 2 q^{27} - q^{29} - 5 q^{33} + q^{35} + 15 q^{37} + 2 q^{39} - 6 q^{41} + q^{43} - q^{45} + 16 q^{47} + 2 q^{49} + q^{51} - 4 q^{53} + 23 q^{55} - 5 q^{57} + 4 q^{59} + 9 q^{61} - 2 q^{63} - q^{65} + 2 q^{67} - 3 q^{69} + 16 q^{71} - 9 q^{73} + 11 q^{75} + 5 q^{77} - 6 q^{79} + 2 q^{81} + 14 q^{83} - 21 q^{85} - q^{87} + 8 q^{89} - 2 q^{91} + 23 q^{95} + 8 q^{97} - 5 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - q^5 - 2 * q^7 + 2 * q^9 - 5 * q^11 + 2 * q^13 - q^15 + q^17 - 5 * q^19 - 2 * q^21 - 3 * q^23 + 11 * q^25 + 2 * q^27 - q^29 - 5 * q^33 + q^35 + 15 * q^37 + 2 * q^39 - 6 * q^41 + q^43 - q^45 + 16 * q^47 + 2 * q^49 + q^51 - 4 * q^53 + 23 * q^55 - 5 * q^57 + 4 * q^59 + 9 * q^61 - 2 * q^63 - q^65 + 2 * q^67 - 3 * q^69 + 16 * q^71 - 9 * q^73 + 11 * q^75 + 5 * q^77 - 6 * q^79 + 2 * q^81 + 14 * q^83 - 21 * q^85 - q^87 + 8 * q^89 - 2 * q^91 + 23 * q^95 + 8 * q^97 - 5 * q^99

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