LMFDB - Embedded newform 1638.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.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:	$13.0794958511$
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			1.1
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.70156 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -2.70156 q^{5} +1.00000 q^{7} -1.00000 q^{8} +2.70156 q^{10} +0.701562 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.70156 q^{17} -0.701562 q^{19} -2.70156 q^{20} -0.701562 q^{22} -4.70156 q^{23} +2.29844 q^{25} -1.00000 q^{26} +1.00000 q^{28} -2.70156 q^{29} -1.00000 q^{32} -2.70156 q^{34} -2.70156 q^{35} +10.7016 q^{37} +0.701562 q^{38} +2.70156 q^{40} -3.40312 q^{41} -10.1047 q^{43} +0.701562 q^{44} +4.70156 q^{46} +8.00000 q^{47} +1.00000 q^{49} -2.29844 q^{50} +1.00000 q^{52} +2.00000 q^{53} -1.89531 q^{55} -1.00000 q^{56} +2.70156 q^{58} +14.8062 q^{59} +1.29844 q^{61} +1.00000 q^{64} -2.70156 q^{65} +5.40312 q^{67} +2.70156 q^{68} +2.70156 q^{70} +8.00000 q^{71} -1.29844 q^{73} -10.7016 q^{74} -0.701562 q^{76} +0.701562 q^{77} +9.40312 q^{79} -2.70156 q^{80} +3.40312 q^{82} +13.4031 q^{83} -7.29844 q^{85} +10.1047 q^{86} -0.701562 q^{88} +8.80625 q^{89} +1.00000 q^{91} -4.70156 q^{92} -8.00000 q^{94} +1.89531 q^{95} -8.80625 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + q^5 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} - 2 q^{8} - q^{10} - 5 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - q^{17} + 5 q^{19} + q^{20} + 5 q^{22} - 3 q^{23} + 11 q^{25} - 2 q^{26} + 2 q^{28} + q^{29} - 2 q^{32} + q^{34} + q^{35} + 15 q^{37} - 5 q^{38} - q^{40} + 6 q^{41} - q^{43} - 5 q^{44} + 3 q^{46} + 16 q^{47} + 2 q^{49} - 11 q^{50} + 2 q^{52} + 4 q^{53} - 23 q^{55} - 2 q^{56} - q^{58} + 4 q^{59} + 9 q^{61} + 2 q^{64} + q^{65} - 2 q^{67} - q^{68} - q^{70} + 16 q^{71} - 9 q^{73} - 15 q^{74} + 5 q^{76} - 5 q^{77} + 6 q^{79} + q^{80} - 6 q^{82} + 14 q^{83} - 21 q^{85} + q^{86} + 5 q^{88} - 8 q^{89} + 2 q^{91} - 3 q^{92} - 16 q^{94} + 23 q^{95} + 8 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + q^5 + 2 * q^7 - 2 * q^8 - q^10 - 5 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 - q^17 + 5 * q^19 + q^20 + 5 * q^22 - 3 * q^23 + 11 * q^25 - 2 * q^26 + 2 * q^28 + q^29 - 2 * q^32 + q^34 + q^35 + 15 * q^37 - 5 * q^38 - q^40 + 6 * q^41 - q^43 - 5 * q^44 + 3 * q^46 + 16 * q^47 + 2 * q^49 - 11 * q^50 + 2 * q^52 + 4 * q^53 - 23 * q^55 - 2 * q^56 - q^58 + 4 * q^59 + 9 * q^61 + 2 * q^64 + q^65 - 2 * q^67 - q^68 - q^70 + 16 * q^71 - 9 * q^73 - 15 * q^74 + 5 * q^76 - 5 * q^77 + 6 * q^79 + q^80 - 6 * q^82 + 14 * q^83 - 21 * q^85 + q^86 + 5 * q^88 - 8 * q^89 + 2 * q^91 - 3 * q^92 - 16 * q^94 + 23 * q^95 + 8 * q^97 - 2 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.70156	−1.20818	−0.604088	−	0.796918i	$-0.706462\pi$
$5$	−2.70156	−1.20818	−0.604088	+	0.796918i	$0.706462\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.70156	0.854309
$11$	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
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.70156	0.655225	0.327613	−	0.944812i	$-0.393756\pi$
$17$	2.70156	0.655225	0.327613	+	0.944812i	$0.393756\pi$
$18$	0	0
$19$	−0.701562	−0.160949	−0.0804747	−	0.996757i	$-0.525644\pi$
$19$	−0.701562	−0.160949	−0.0804747	+	0.996757i	$0.525644\pi$
$20$	−2.70156	−0.604088
$21$	0	0
$22$	−0.701562	−0.149574
$23$	−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
$26$	−1.00000	−0.196116
$27$	0	0
$28$	1.00000	0.188982
$29$	−2.70156	−0.501667	−0.250834	−	0.968030i	$-0.580705\pi$
$29$	−2.70156	−0.501667	−0.250834	+	0.968030i	$0.580705\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.70156	−0.463314
$35$	−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	$0.157768\pi$
$38$	0.701562	0.113808
$39$	0	0
$40$	2.70156	0.427154
$41$	−3.40312	−0.531479	−0.265739	−	0.964045i	$-0.585616\pi$
$41$	−3.40312	−0.531479	−0.265739	+	0.964045i	$0.585616\pi$
$42$	0	0
$43$	−10.1047	−1.54095	−0.770475	−	0.637470i	$-0.779981\pi$
$43$	−10.1047	−1.54095	−0.770475	+	0.637470i	$0.779981\pi$
$44$	0.701562	0.105764
$45$	0	0
$46$	4.70156	0.693208
$47$	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
$50$	−2.29844	−0.325048
$51$	0	0
$52$	1.00000	0.138675
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−1.89531	−0.255564
$56$	−1.00000	−0.133631
$57$	0	0
$58$	2.70156	0.354732
$59$	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
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.70156	−0.335088
$66$	0	0
$67$	5.40312	0.660097	0.330048	−	0.943964i	$-0.392935\pi$
$67$	5.40312	0.660097	0.330048	+	0.943964i	$0.392935\pi$
$68$	2.70156	0.327613
$69$	0	0
$70$	2.70156	0.322898
$71$	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	$0.524210\pi$
$74$	−10.7016	−1.24403
$75$	0	0
$76$	−0.701562	−0.0804747
$77$	0.701562	0.0799504
$78$	0	0
$79$	9.40312	1.05793	0.528967	−	0.848642i	$-0.322579\pi$
$79$	9.40312	1.05793	0.528967	+	0.848642i	$0.322579\pi$
$80$	−2.70156	−0.302044
$81$	0	0
$82$	3.40312	0.375812
$83$	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
$86$	10.1047	1.08962
$87$	0	0
$88$	−0.701562	−0.0747868
$89$	8.80625	0.933460	0.466730	−	0.884400i	$-0.345432\pi$
$89$	8.80625	0.933460	0.466730	+	0.884400i	$0.345432\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−4.70156	−0.490172
$93$	0	0
$94$	−8.00000	−0.825137
$95$	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	$0.647532\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	2.29844	0.229844
$101$	3.40312	0.338624	0.169312	−	0.985563i	$-0.445846\pi$
$101$	3.40312	0.338624	0.169312	+	0.985563i	$0.445846\pi$
$102$	0	0
$103$	−3.29844	−0.325005	−0.162502	−	0.986708i	$-0.551957\pi$
$103$	−3.29844	−0.325005	−0.162502	+	0.986708i	$0.551957\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−2.00000	−0.194257
$107$	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	$0.353092\pi$
$110$	1.89531	0.180711
$111$	0	0
$112$	1.00000	0.0944911
$113$	−4.80625	−0.452134	−0.226067	−	0.974112i	$-0.572587\pi$
$113$	−4.80625	−0.452134	−0.226067	+	0.974112i	$0.572587\pi$
$114$	0	0
$115$	12.7016	1.18443
$116$	−2.70156	−0.250834
$117$	0	0
$118$	−14.8062	−1.36303
$119$	2.70156	0.247652
$120$	0	0
$121$	−10.5078	−0.955256
$122$	−1.29844	−0.117555
$123$	0	0
$124$	0	0
$125$	7.29844	0.652792
$126$	0	0
$127$	−6.59688	−0.585378	−0.292689	−	0.956208i	$-0.594550\pi$
$127$	−6.59688	−0.585378	−0.292689	+	0.956208i	$0.594550\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	2.70156	0.236943
$131$	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
$134$	−5.40312	−0.466759
$135$	0	0
$136$	−2.70156	−0.231657
$137$	18.7016	1.59778	0.798891	−	0.601476i	$-0.205420\pi$
$137$	18.7016	1.59778	0.798891	+	0.601476i	$0.205420\pi$
$138$	0	0
$139$	6.80625	0.577298	0.288649	−	0.957435i	$-0.406794\pi$
$139$	6.80625	0.577298	0.288649	+	0.957435i	$0.406794\pi$
$140$	−2.70156	−0.228324
$141$	0	0
$142$	−8.00000	−0.671345
$143$	0.701562	0.0586676
$144$	0	0
$145$	7.29844	0.606102
$146$	1.29844	0.107459
$147$	0	0
$148$	10.7016	0.879663
$149$	−15.4031	−1.26187	−0.630937	−	0.775834i	$-0.717329\pi$
$149$	−15.4031	−1.26187	−0.630937	+	0.775834i	$0.717329\pi$
$150$	0	0
$151$	−4.70156	−0.382608	−0.191304	−	0.981531i	$-0.561272\pi$
$151$	−4.70156	−0.382608	−0.191304	+	0.981531i	$0.561272\pi$
$152$	0.701562	0.0569042
$153$	0	0
$154$	−0.701562	−0.0565335
$155$	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	$0.203628\pi$
$158$	−9.40312	−0.748072
$159$	0	0
$160$	2.70156	0.213577
$161$	−4.70156	−0.370535
$162$	0	0
$163$	5.40312	0.423205	0.211603	−	0.977356i	$-0.432132\pi$
$163$	5.40312	0.423205	0.211603	+	0.977356i	$0.432132\pi$
$164$	−3.40312	−0.265739
$165$	0	0
$166$	−13.4031	−1.04028
$167$	−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
$170$	7.29844	0.559765
$171$	0	0
$172$	−10.1047	−0.770475
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	2.29844	0.173746
$176$	0.701562	0.0528822
$177$	0	0
$178$	−8.80625	−0.660056
$179$	−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	$0.393867\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	4.70156	0.346604
$185$	−28.9109	−2.12557
$186$	0	0
$187$	1.89531	0.138599
$188$	8.00000	0.583460
$189$	0	0
$190$	−1.89531	−0.137501
$191$	−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	$0.365386\pi$
$194$	8.80625	0.632252
$195$	0	0
$196$	1.00000	0.0714286
$197$	3.40312	0.242463	0.121231	−	0.992624i	$-0.461316\pi$
$197$	3.40312	0.242463	0.121231	+	0.992624i	$0.461316\pi$
$198$	0	0
$199$	22.1047	1.56696	0.783480	−	0.621417i	$-0.213443\pi$
$199$	22.1047	1.56696	0.783480	+	0.621417i	$0.213443\pi$
$200$	−2.29844	−0.162524
$201$	0	0
$202$	−3.40312	−0.239443
$203$	−2.70156	−0.189612
$204$	0	0
$205$	9.19375	0.642119
$206$	3.29844	0.229813
$207$	0	0
$208$	1.00000	0.0693375
$209$	−0.492189	−0.0340455
$210$	0	0
$211$	−24.7016	−1.70053	−0.850263	−	0.526358i	$-0.823557\pi$
$211$	−24.7016	−1.70053	−0.850263	+	0.526358i	$0.823557\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−5.40312	−0.369350
$215$	27.2984	1.86174
$216$	0	0
$217$	0	0
$218$	−9.29844	−0.629770
$219$	0	0
$220$	−1.89531	−0.127782
$221$	2.70156	0.181727
$222$	0	0
$223$	9.40312	0.629680	0.314840	−	0.949145i	$-0.398049\pi$
$223$	9.40312	0.629680	0.314840	+	0.949145i	$0.398049\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	4.80625	0.319707
$227$	−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	$0.436476\pi$
$230$	−12.7016	−0.837516
$231$	0	0
$232$	2.70156	0.177366
$233$	18.2094	1.19294	0.596468	−	0.802637i	$-0.296570\pi$
$233$	18.2094	1.19294	0.596468	+	0.802637i	$0.296570\pi$
$234$	0	0
$235$	−21.6125	−1.40984
$236$	14.8062	0.963805
$237$	0	0
$238$	−2.70156	−0.175116
$239$	−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	$0.794614\pi$
$242$	10.5078	0.675468
$243$	0	0
$244$	1.29844	0.0831240
$245$	−2.70156	−0.172596
$246$	0	0
$247$	−0.701562	−0.0446393
$248$	0	0
$249$	0	0
$250$	−7.29844	−0.461594
$251$	−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
$254$	6.59688	0.413925
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	10.7016	0.664963
$260$	−2.70156	−0.167544
$261$	0	0
$262$	−7.29844	−0.450899
$263$	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
$266$	0.701562	0.0430155
$267$	0	0
$268$	5.40312	0.330048
$269$	4.80625	0.293042	0.146521	−	0.989208i	$-0.453192\pi$
$269$	4.80625	0.293042	0.146521	+	0.989208i	$0.453192\pi$
$270$	0	0
$271$	12.2094	0.741667	0.370833	−	0.928699i	$-0.379072\pi$
$271$	12.2094	0.741667	0.370833	+	0.928699i	$0.379072\pi$
$272$	2.70156	0.163806
$273$	0	0
$274$	−18.7016	−1.12980
$275$	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	$0.188604\pi$
$278$	−6.80625	−0.408212
$279$	0	0
$280$	2.70156	0.161449
$281$	−12.8062	−0.763957	−0.381978	−	0.924171i	$-0.624757\pi$
$281$	−12.8062	−0.763957	−0.381978	+	0.924171i	$0.624757\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−0.701562	−0.0414842
$287$	−3.40312	−0.200880
$288$	0	0
$289$	−9.70156	−0.570680
$290$	−7.29844	−0.428579
$291$	0	0
$292$	−1.29844	−0.0759853
$293$	12.8062	0.748149	0.374075	−	0.927399i	$-0.377960\pi$
$293$	12.8062	0.748149	0.374075	+	0.927399i	$0.377960\pi$
$294$	0	0
$295$	−40.0000	−2.32889
$296$	−10.7016	−0.622016
$297$	0	0
$298$	15.4031	0.892279
$299$	−4.70156	−0.271898
$300$	0	0
$301$	−10.1047	−0.582424
$302$	4.70156	0.270544
$303$	0	0
$304$	−0.701562	−0.0402373
$305$	−3.50781	−0.200857
$306$	0	0
$307$	−6.80625	−0.388453	−0.194227	−	0.980957i	$-0.562220\pi$
$307$	−6.80625	−0.388453	−0.194227	+	0.980957i	$0.562220\pi$
$308$	0.701562	0.0399752
$309$	0	0
$310$	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	$0.284007\pi$
$314$	−20.1047	−1.13457
$315$	0	0
$316$	9.40312	0.528967
$317$	−7.40312	−0.415801	−0.207900	−	0.978150i	$-0.566663\pi$
$317$	−7.40312	−0.415801	−0.207900	+	0.978150i	$0.566663\pi$
$318$	0	0
$319$	−1.89531	−0.106117
$320$	−2.70156	−0.151022
$321$	0	0
$322$	4.70156	0.262008
$323$	−1.89531	−0.105458
$324$	0	0
$325$	2.29844	0.127494
$326$	−5.40312	−0.299251
$327$	0	0
$328$	3.40312	0.187906
$329$	8.00000	0.441054
$330$	0	0
$331$	32.2094	1.77039	0.885194	−	0.465223i	$-0.154026\pi$
$331$	32.2094	1.77039	0.885194	+	0.465223i	$0.154026\pi$
$332$	13.4031	0.735592
$333$	0	0
$334$	3.29844	0.180482
$335$	−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	$0.797136\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	−7.29844	−0.395813
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	10.1047	0.544808
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−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	$0.203272\pi$
$350$	−2.29844	−0.122857
$351$	0	0
$352$	−0.701562	−0.0373934
$353$	−20.8062	−1.10740	−0.553702	−	0.832715i	$-0.686785\pi$
$353$	−20.8062	−1.10740	−0.553702	+	0.832715i	$0.686785\pi$
$354$	0	0
$355$	−21.6125	−1.14707
$356$	8.80625	0.466730
$357$	0	0
$358$	14.8062	0.782535
$359$	−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
$362$	−8.80625	−0.462846
$363$	0	0
$364$	1.00000	0.0524142
$365$	3.50781	0.183607
$366$	0	0
$367$	−29.6125	−1.54576	−0.772880	−	0.634552i	$-0.781184\pi$
$367$	−29.6125	−1.54576	−0.772880	+	0.634552i	$0.781184\pi$
$368$	−4.70156	−0.245086
$369$	0	0
$370$	28.9109	1.50301
$371$	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	$0.667523\pi$
$374$	−1.89531	−0.0980043
$375$	0	0
$376$	−8.00000	−0.412568
$377$	−2.70156	−0.139138
$378$	0	0
$379$	−17.6125	−0.904693	−0.452347	−	0.891842i	$-0.649413\pi$
$379$	−17.6125	−0.904693	−0.452347	+	0.891842i	$0.649413\pi$
$380$	1.89531	0.0972275
$381$	0	0
$382$	12.7016	0.649868
$383$	−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
$386$	−11.4031	−0.580404
$387$	0	0
$388$	−8.80625	−0.447070
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	−12.7016	−0.642346
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−3.40312	−0.171447
$395$	−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	$0.493559\pi$
$398$	−22.1047	−1.10801
$399$	0	0
$400$	2.29844	0.114922
$401$	−4.80625	−0.240013	−0.120006	−	0.992773i	$-0.538291\pi$
$401$	−4.80625	−0.240013	−0.120006	+	0.992773i	$0.538291\pi$
$402$	0	0
$403$	0	0
$404$	3.40312	0.169312
$405$	0	0
$406$	2.70156	0.134076
$407$	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	$0.458183\pi$
$410$	−9.19375	−0.454047
$411$	0	0
$412$	−3.29844	−0.162502
$413$	14.8062	0.728568
$414$	0	0
$415$	−36.2094	−1.77745
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	0.492189	0.0240738
$419$	−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	$0.341388\pi$
$422$	24.7016	1.20245
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	6.20937	0.301199
$426$	0	0
$427$	1.29844	0.0628358
$428$	5.40312	0.261170
$429$	0	0
$430$	−27.2984	−1.31645
$431$	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
$435$	0	0
$436$	9.29844	0.445314
$437$	3.29844	0.157786
$438$	0	0
$439$	−0.492189	−0.0234909	−0.0117455	−	0.999931i	$-0.503739\pi$
$439$	−0.492189	−0.0234909	−0.0117455	+	0.999931i	$0.503739\pi$
$440$	1.89531	0.0903555
$441$	0	0
$442$	−2.70156	−0.128500
$443$	−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
$446$	−9.40312	−0.445251
$447$	0	0
$448$	1.00000	0.0472456
$449$	7.89531	0.372603	0.186301	−	0.982493i	$-0.440350\pi$
$449$	7.89531	0.372603	0.186301	+	0.982493i	$0.440350\pi$
$450$	0	0
$451$	−2.38750	−0.112423
$452$	−4.80625	−0.226067
$453$	0	0
$454$	21.4031	1.00450
$455$	−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	$0.495556\pi$
$458$	−6.00000	−0.280362
$459$	0	0
$460$	12.7016	0.592213
$461$	20.3141	0.946120	0.473060	−	0.881030i	$-0.343149\pi$
$461$	20.3141	0.946120	0.473060	+	0.881030i	$0.343149\pi$
$462$	0	0
$463$	−34.3141	−1.59471	−0.797355	−	0.603511i	$-0.793768\pi$
$463$	−34.3141	−1.59471	−0.797355	+	0.603511i	$0.793768\pi$
$464$	−2.70156	−0.125417
$465$	0	0
$466$	−18.2094	−0.843533
$467$	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
$470$	21.6125	0.996910
$471$	0	0
$472$	−14.8062	−0.681513
$473$	−7.08907	−0.325956
$474$	0	0
$475$	−1.61250	−0.0739864
$476$	2.70156	0.123826
$477$	0	0
$478$	16.0000	0.731823
$479$	−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
$482$	24.8062	1.12989
$483$	0	0
$484$	−10.5078	−0.477628
$485$	23.7906	1.08028
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−1.29844	−0.0587775
$489$	0	0
$490$	2.70156	0.122044
$491$	−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
$494$	0.701562	0.0315648
$495$	0	0
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	−1.19375	−0.0534397	−0.0267198	−	0.999643i	$-0.508506\pi$
$499$	−1.19375	−0.0534397	−0.0267198	+	0.999643i	$0.508506\pi$
$500$	7.29844	0.326396
$501$	0	0
$502$	3.50781	0.156561
$503$	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
$506$	3.29844	0.146633
$507$	0	0
$508$	−6.59688	−0.292689
$509$	−38.9109	−1.72470	−0.862348	−	0.506315i	$-0.831007\pi$
$509$	−38.9109	−1.72470	−0.862348	+	0.506315i	$0.831007\pi$
$510$	0	0
$511$	−1.29844	−0.0574395
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−6.00000	−0.264649
$515$	8.91093	0.392663
$516$	0	0
$517$	5.61250	0.246837
$518$	−10.7016	−0.470200
$519$	0	0
$520$	2.70156	0.118471
$521$	1.29844	0.0568856	0.0284428	−	0.999595i	$-0.490945\pi$
$521$	1.29844	0.0568856	0.0284428	+	0.999595i	$0.490945\pi$
$522$	0	0
$523$	−33.6125	−1.46977	−0.734886	−	0.678191i	$-0.762764\pi$
$523$	−33.6125	−1.46977	−0.734886	+	0.678191i	$0.762764\pi$
$524$	7.29844	0.318834
$525$	0	0
$526$	−26.8062	−1.16881
$527$	0	0
$528$	0	0
$529$	−0.895314	−0.0389267
$530$	5.40312	0.234697
$531$	0	0
$532$	−0.701562	−0.0304166
$533$	−3.40312	−0.147406
$534$	0	0
$535$	−14.5969	−0.631078
$536$	−5.40312	−0.233379
$537$	0	0
$538$	−4.80625	−0.207212
$539$	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	$0.546016\pi$
$542$	−12.2094	−0.524437
$543$	0	0
$544$	−2.70156	−0.115829
$545$	−25.1203	−1.07604
$546$	0	0
$547$	9.61250	0.411001	0.205500	−	0.978657i	$-0.434118\pi$
$547$	9.61250	0.411001	0.205500	+	0.978657i	$0.434118\pi$
$548$	18.7016	0.798891
$549$	0	0
$550$	−1.61250	−0.0687571
$551$	1.89531	0.0807431
$552$	0	0
$553$	9.40312	0.399862
$554$	−27.6125	−1.17314
$555$	0	0
$556$	6.80625	0.288649
$557$	24.5969	1.04220	0.521102	−	0.853495i	$-0.325521\pi$
$557$	24.5969	1.04220	0.521102	+	0.853495i	$0.325521\pi$
$558$	0	0
$559$	−10.1047	−0.427383
$560$	−2.70156	−0.114162
$561$	0	0
$562$	12.8062	0.540199
$563$	−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
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−8.00000	−0.335673
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	−1.19375	−0.0499569	−0.0249785	−	0.999688i	$-0.507952\pi$
$571$	−1.19375	−0.0499569	−0.0249785	+	0.999688i	$0.507952\pi$
$572$	0.701562	0.0293338
$573$	0	0
$574$	3.40312	0.142044
$575$	−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	$0.558679\pi$
$578$	9.70156	0.403532
$579$	0	0
$580$	7.29844	0.303051
$581$	13.4031	0.556055
$582$	0	0
$583$	1.40312	0.0581115
$584$	1.29844	0.0537297
$585$	0	0
$586$	−12.8062	−0.529021
$587$	−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
$590$	40.0000	1.64677
$591$	0	0
$592$	10.7016	0.439831
$593$	16.8062	0.690150	0.345075	−	0.938575i	$-0.387853\pi$
$593$	16.8062	0.690150	0.345075	+	0.938575i	$0.387853\pi$
$594$	0	0
$595$	−7.29844	−0.299207
$596$	−15.4031	−0.630937
$597$	0	0
$598$	4.70156	0.192261
$599$	−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	$0.601721\pi$
$602$	10.1047	0.411836
$603$	0	0
$604$	−4.70156	−0.191304
$605$	28.3875	1.15412
$606$	0	0
$607$	7.50781	0.304733	0.152366	−	0.988324i	$-0.451311\pi$
$607$	7.50781	0.304733	0.152366	+	0.988324i	$0.451311\pi$
$608$	0.701562	0.0284521
$609$	0	0
$610$	3.50781	0.142027
$611$	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	$0.212252\pi$
$614$	6.80625	0.274678
$615$	0	0
$616$	−0.701562	−0.0282667
$617$	30.9109	1.24443	0.622214	−	0.782847i	$-0.286233\pi$
$617$	30.9109	1.24443	0.622214	+	0.782847i	$0.286233\pi$
$618$	0	0
$619$	7.29844	0.293349	0.146674	−	0.989185i	$-0.453143\pi$
$619$	7.29844	0.293349	0.146674	+	0.989185i	$0.453143\pi$
$620$	0	0
$621$	0	0
$622$	14.5969	0.585281
$623$	8.80625	0.352815
$624$	0	0
$625$	−31.2094	−1.24837
$626$	−22.2094	−0.887665
$627$	0	0
$628$	20.1047	0.802264
$629$	28.9109	1.15275
$630$	0	0
$631$	−7.50781	−0.298881	−0.149441	−	0.988771i	$-0.547747\pi$
$631$	−7.50781	−0.298881	−0.149441	+	0.988771i	$0.547747\pi$
$632$	−9.40312	−0.374036
$633$	0	0
$634$	7.40312	0.294016
$635$	17.8219	0.707239
$636$	0	0
$637$	1.00000	0.0396214
$638$	1.89531	0.0750362
$639$	0	0
$640$	2.70156	0.106789
$641$	48.8062	1.92773	0.963865	−	0.266390i	$-0.0858308\pi$
$641$	48.8062	1.92773	0.963865	+	0.266390i	$0.0858308\pi$
$642$	0	0
$643$	46.3141	1.82645	0.913224	−	0.407458i	$-0.133585\pi$
$643$	46.3141	1.82645	0.913224	+	0.407458i	$0.133585\pi$
$644$	−4.70156	−0.185268
$645$	0	0
$646$	1.89531	0.0745701
$647$	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
$650$	−2.29844	−0.0901522
$651$	0	0
$652$	5.40312	0.211603
$653$	9.50781	0.372069	0.186035	−	0.982543i	$-0.440436\pi$
$653$	9.50781	0.372069	0.186035	+	0.982543i	$0.440436\pi$
$654$	0	0
$655$	−19.7172	−0.770414
$656$	−3.40312	−0.132870
$657$	0	0
$658$	−8.00000	−0.311872
$659$	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	$0.937086\pi$
$662$	−32.2094	−1.25185
$663$	0	0
$664$	−13.4031	−0.520142
$665$	1.89531	0.0734971
$666$	0	0
$667$	12.7016	0.491806
$668$	−3.29844	−0.127620
$669$	0	0
$670$	14.5969	0.563927
$671$	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	$0.190019\pi$
$674$	29.5078	1.13660
$675$	0	0
$676$	1.00000	0.0384615
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	−8.80625	−0.337953
$680$	7.29844	0.279882
$681$	0	0
$682$	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
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−10.1047	−0.385238
$689$	2.00000	0.0761939
$690$	0	0
$691$	49.6125	1.88735	0.943674	−	0.330876i	$-0.107344\pi$
$691$	49.6125	1.88735	0.943674	+	0.330876i	$0.107344\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	4.00000	0.151838
$695$	−18.3875	−0.697478
$696$	0	0
$697$	−9.19375	−0.348238
$698$	−30.0000	−1.13552
$699$	0	0
$700$	2.29844	0.0868728
$701$	−3.19375	−0.120626	−0.0603132	−	0.998180i	$-0.519210\pi$
$701$	−3.19375	−0.120626	−0.0603132	+	0.998180i	$0.519210\pi$
$702$	0	0
$703$	−7.50781	−0.283162
$704$	0.701562	0.0264411
$705$	0	0
$706$	20.8062	0.783053
$707$	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	$0.0792402\pi$
$710$	21.6125	0.811103
$711$	0	0
$712$	−8.80625	−0.330028
$713$	0	0
$714$	0	0
$715$	−1.89531	−0.0708807
$716$	−14.8062	−0.553335
$717$	0	0
$718$	26.8062	1.00040
$719$	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
$722$	18.5078	0.688789
$723$	0	0
$724$	8.80625	0.327282
$725$	−6.20937	−0.230610
$726$	0	0
$727$	−22.1047	−0.819817	−0.409909	−	0.912127i	$-0.634439\pi$
$727$	−22.1047	−0.819817	−0.409909	+	0.912127i	$0.634439\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	−3.50781	−0.129830
$731$	−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	$0.375793\pi$
$734$	29.6125	1.09302
$735$	0	0
$736$	4.70156	0.173302
$737$	3.79063	0.139630
$738$	0	0
$739$	40.2094	1.47913	0.739563	−	0.673088i	$-0.235032\pi$
$739$	40.2094	1.47913	0.739563	+	0.673088i	$0.235032\pi$
$740$	−28.9109	−1.06279
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	−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
$746$	19.4031	0.710399
$747$	0	0
$748$	1.89531	0.0692995
$749$	5.40312	0.197426
$750$	0	0
$751$	34.8062	1.27010	0.635049	−	0.772472i	$-0.280980\pi$
$751$	34.8062	1.27010	0.635049	+	0.772472i	$0.280980\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	2.70156	0.0983851
$755$	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	$0.313562\pi$
$758$	17.6125	0.639715
$759$	0	0
$760$	−1.89531	−0.0687503
$761$	−32.5969	−1.18164	−0.590818	−	0.806805i	$-0.701195\pi$
$761$	−32.5969	−1.18164	−0.590818	+	0.806805i	$0.701195\pi$
$762$	0	0
$763$	9.29844	0.336626
$764$	−12.7016	−0.459526
$765$	0	0
$766$	16.9109	0.611017
$767$	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	$0.129845\pi$
$770$	1.89531	0.0683024
$771$	0	0
$772$	11.4031	0.410407
$773$	28.3141	1.01839	0.509193	−	0.860652i	$-0.329944\pi$
$773$	28.3141	1.01839	0.509193	+	0.860652i	$0.329944\pi$
$774$	0	0
$775$	0	0
$776$	8.80625	0.316126
$777$	0	0
$778$	14.0000	0.501924
$779$	2.38750	0.0855412
$780$	0	0
$781$	5.61250	0.200831
$782$	12.7016	0.454207
$783$	0	0
$784$	1.00000	0.0357143
$785$	−54.3141	−1.93855
$786$	0	0
$787$	20.9109	0.745394	0.372697	−	0.927953i	$-0.378433\pi$
$787$	20.9109	0.745394	0.372697	+	0.927953i	$0.378433\pi$
$788$	3.40312	0.121231
$789$	0	0
$790$	25.4031	0.903803
$791$	−4.80625	−0.170891
$792$	0	0
$793$	1.29844	0.0461089
$794$	−0.806248	−0.0286127
$795$	0	0
$796$	22.1047	0.783480
$797$	40.5969	1.43802	0.719008	−	0.695002i	$-0.244597\pi$
$797$	40.5969	1.43802	0.719008	+	0.695002i	$0.244597\pi$
$798$	0	0
$799$	21.6125	0.764595
$800$	−2.29844	−0.0812621
$801$	0	0
$802$	4.80625	0.169715
$803$	−0.910935	−0.0321462
$804$	0	0
$805$	12.7016	0.447671
$806$	0	0
$807$	0	0
$808$	−3.40312	−0.119721
$809$	11.6125	0.408274	0.204137	−	0.978942i	$-0.434561\pi$
$809$	11.6125	0.408274	0.204137	+	0.978942i	$0.434561\pi$
$810$	0	0
$811$	−21.8953	−0.768848	−0.384424	−	0.923157i	$-0.625600\pi$
$811$	−21.8953	−0.768848	−0.384424	+	0.923157i	$0.625600\pi$
$812$	−2.70156	−0.0948062
$813$	0	0
$814$	−7.50781	−0.263149
$815$	−14.5969	−0.511306
$816$	0	0
$817$	7.08907	0.248015
$818$	−5.29844	−0.185256
$819$	0	0
$820$	9.19375	0.321060
$821$	−28.5969	−0.998038	−0.499019	−	0.866591i	$-0.666306\pi$
$821$	−28.5969	−0.998038	−0.499019	+	0.866591i	$0.666306\pi$
$822$	0	0
$823$	5.19375	0.181043	0.0905214	−	0.995895i	$-0.471147\pi$
$823$	5.19375	0.181043	0.0905214	+	0.995895i	$0.471147\pi$
$824$	3.29844	0.114907
$825$	0	0
$826$	−14.8062	−0.515175
$827$	−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	$0.717199\pi$
$830$	36.2094	1.25685
$831$	0	0
$832$	1.00000	0.0346688
$833$	2.70156	0.0936036
$834$	0	0
$835$	8.91093	0.308376
$836$	−0.492189	−0.0170227
$837$	0	0
$838$	34.1047	1.17813
$839$	−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
$842$	−19.6125	−0.675891
$843$	0	0
$844$	−24.7016	−0.850263
$845$	−2.70156	−0.0929366
$846$	0	0
$847$	−10.5078	−0.361053
$848$	2.00000	0.0686803
$849$	0	0
$850$	−6.20937	−0.212980
$851$	−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	$0.624151\pi$
$854$	−1.29844	−0.0444316
$855$	0	0
$856$	−5.40312	−0.184675
$857$	16.8062	0.574091	0.287045	−	0.957917i	$-0.407327\pi$
$857$	16.8062	0.574091	0.287045	+	0.957917i	$0.407327\pi$
$858$	0	0
$859$	38.8062	1.32405	0.662026	−	0.749481i	$-0.269697\pi$
$859$	38.8062	1.32405	0.662026	+	0.749481i	$0.269697\pi$
$860$	27.2984	0.930869
$861$	0	0
$862$	−12.2094	−0.415853
$863$	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
$866$	−14.2094	−0.482854
$867$	0	0
$868$	0	0
$869$	6.59688	0.223784
$870$	0	0
$871$	5.40312	0.183078
$872$	−9.29844	−0.314885
$873$	0	0
$874$	−3.29844	−0.111571
$875$	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	$0.497917\pi$
$878$	0.492189	0.0166106
$879$	0	0
$880$	−1.89531	−0.0638910
$881$	8.31406	0.280108	0.140054	−	0.990144i	$-0.455272\pi$
$881$	8.31406	0.280108	0.140054	+	0.990144i	$0.455272\pi$
$882$	0	0
$883$	13.8953	0.467615	0.233807	−	0.972283i	$-0.424882\pi$
$883$	13.8953	0.467615	0.233807	+	0.972283i	$0.424882\pi$
$884$	2.70156	0.0908634
$885$	0	0
$886$	0.209373	0.00703401
$887$	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
$890$	23.7906	0.797464
$891$	0	0
$892$	9.40312	0.314840
$893$	−5.61250	−0.187815
$894$	0	0
$895$	40.0000	1.33705
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−7.89531	−0.263470
$899$	0	0
$900$	0	0
$901$	5.40312	0.180004
$902$	2.38750	0.0794952
$903$	0	0
$904$	4.80625	0.159853
$905$	−23.7906	−0.790827
$906$	0	0
$907$	−25.6125	−0.850449	−0.425225	−	0.905088i	$-0.639805\pi$
$907$	−25.6125	−0.850449	−0.425225	+	0.905088i	$0.639805\pi$
$908$	−21.4031	−0.710288
$909$	0	0
$910$	2.70156	0.0895559
$911$	−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
$914$	−0.596876	−0.0197429
$915$	0	0
$916$	6.00000	0.198246
$917$	7.29844	0.241016
$918$	0	0
$919$	−14.5969	−0.481507	−0.240753	−	0.970586i	$-0.577394\pi$
$919$	−14.5969	−0.481507	−0.240753	+	0.970586i	$0.577394\pi$
$920$	−12.7016	−0.418758
$921$	0	0
$922$	−20.3141	−0.669008
$923$	8.00000	0.263323
$924$	0	0
$925$	24.5969	0.808740
$926$	34.3141	1.12763
$927$	0	0
$928$	2.70156	0.0886831
$929$	−33.0156	−1.08321	−0.541604	−	0.840634i	$-0.682183\pi$
$929$	−33.0156	−1.08321	−0.541604	+	0.840634i	$0.682183\pi$
$930$	0	0
$931$	−0.701562	−0.0229928
$932$	18.2094	0.596468
$933$	0	0
$934$	−4.49219	−0.146989
$935$	−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	$0.344063\pi$
$938$	−5.40312	−0.176418
$939$	0	0
$940$	−21.6125	−0.704922
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	14.8062	0.481902
$945$	0	0
$946$	7.08907	0.230485
$947$	−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
$950$	1.61250	0.0523163
$951$	0	0
$952$	−2.70156	−0.0875581
$953$	39.8219	1.28996	0.644978	−	0.764201i	$-0.276866\pi$
$953$	39.8219	1.28996	0.644978	+	0.764201i	$0.276866\pi$
$954$	0	0
$955$	34.3141	1.11038
$956$	−16.0000	−0.517477
$957$	0	0
$958$	7.50781	0.242566
$959$	18.7016	0.603905
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−10.7016	−0.345032
$963$	0	0
$964$	−24.8062	−0.798955
$965$	−30.8062	−0.991688
$966$	0	0
$967$	−51.7172	−1.66311	−0.831556	−	0.555441i	$-0.812550\pi$
$967$	−51.7172	−1.66311	−0.831556	+	0.555441i	$0.812550\pi$
$968$	10.5078	0.337734
$969$	0	0
$970$	−23.7906	−0.763871
$971$	−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
$974$	8.00000	0.256337
$975$	0	0
$976$	1.29844	0.0415620
$977$	41.7172	1.33465	0.667325	−	0.744766i	$-0.267439\pi$
$977$	41.7172	1.33465	0.667325	+	0.744766i	$0.267439\pi$
$978$	0	0
$979$	6.17813	0.197454
$980$	−2.70156	−0.0862982
$981$	0	0
$982$	26.5969	0.848740
$983$	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
$986$	7.29844	0.232430
$987$	0	0
$988$	−0.701562	−0.0223197
$989$	47.5078	1.51066
$990$	0	0
$991$	−38.5969	−1.22607	−0.613035	−	0.790056i	$-0.710052\pi$
$991$	−38.5969	−1.22607	−0.613035	+	0.790056i	$0.710052\pi$
$992$	0	0
$993$	0	0
$994$	−8.00000	−0.253745
$995$	−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	$0.565001\pi$
$998$	1.19375	0.0377875
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.w.1.1		2
3.2	odd	2		546.2.a.i.1.2	✓	2
12.11	even	2		4368.2.a.bg.1.2		2
21.20	even	2		3822.2.a.bt.1.1		2
39.38	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	3.2	odd	2
1638.2.a.w.1.1		2	1.1	even	1	trivial
3822.2.a.bt.1.1		2	21.20	even	2
4368.2.a.bg.1.2		2	12.11	even	2
7098.2.a.bh.1.1		2	39.38	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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.70156 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.70156 q^{5} +1.00000 q^{7} -1.00000 q^{8} +2.70156 q^{10} +0.701562 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.70156 q^{17} -0.701562 q^{19} -2.70156 q^{20} -0.701562 q^{22} -4.70156 q^{23} +2.29844 q^{25} -1.00000 q^{26} +1.00000 q^{28} -2.70156 q^{29} -1.00000 q^{32} -2.70156 q^{34} -2.70156 q^{35} +10.7016 q^{37} +0.701562 q^{38} +2.70156 q^{40} -3.40312 q^{41} -10.1047 q^{43} +0.701562 q^{44} +4.70156 q^{46} +8.00000 q^{47} +1.00000 q^{49} -2.29844 q^{50} +1.00000 q^{52} +2.00000 q^{53} -1.89531 q^{55} -1.00000 q^{56} +2.70156 q^{58} +14.8062 q^{59} +1.29844 q^{61} +1.00000 q^{64} -2.70156 q^{65} +5.40312 q^{67} +2.70156 q^{68} +2.70156 q^{70} +8.00000 q^{71} -1.29844 q^{73} -10.7016 q^{74} -0.701562 q^{76} +0.701562 q^{77} +9.40312 q^{79} -2.70156 q^{80} +3.40312 q^{82} +13.4031 q^{83} -7.29844 q^{85} +10.1047 q^{86} -0.701562 q^{88} +8.80625 q^{89} +1.00000 q^{91} -4.70156 q^{92} -8.00000 q^{94} +1.89531 q^{95} -8.80625 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + q^5 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} - 2 q^{8} - q^{10} - 5 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - q^{17} + 5 q^{19} + q^{20} + 5 q^{22} - 3 q^{23} + 11 q^{25} - 2 q^{26} + 2 q^{28} + q^{29} - 2 q^{32} + q^{34} + q^{35} + 15 q^{37} - 5 q^{38} - q^{40} + 6 q^{41} - q^{43} - 5 q^{44} + 3 q^{46} + 16 q^{47} + 2 q^{49} - 11 q^{50} + 2 q^{52} + 4 q^{53} - 23 q^{55} - 2 q^{56} - q^{58} + 4 q^{59} + 9 q^{61} + 2 q^{64} + q^{65} - 2 q^{67} - q^{68} - q^{70} + 16 q^{71} - 9 q^{73} - 15 q^{74} + 5 q^{76} - 5 q^{77} + 6 q^{79} + q^{80} - 6 q^{82} + 14 q^{83} - 21 q^{85} + q^{86} + 5 q^{88} - 8 q^{89} + 2 q^{91} - 3 q^{92} - 16 q^{94} + 23 q^{95} + 8 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + q^5 + 2 * q^7 - 2 * q^8 - q^10 - 5 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 - q^17 + 5 * q^19 + q^20 + 5 * q^22 - 3 * q^23 + 11 * q^25 - 2 * q^26 + 2 * q^28 + q^29 - 2 * q^32 + q^34 + q^35 + 15 * q^37 - 5 * q^38 - q^40 + 6 * q^41 - q^43 - 5 * q^44 + 3 * q^46 + 16 * q^47 + 2 * q^49 - 11 * q^50 + 2 * q^52 + 4 * q^53 - 23 * q^55 - 2 * q^56 - q^58 + 4 * q^59 + 9 * q^61 + 2 * q^64 + q^65 - 2 * q^67 - q^68 - q^70 + 16 * q^71 - 9 * q^73 - 15 * q^74 + 5 * q^76 - 5 * q^77 + 6 * q^79 + q^80 - 6 * q^82 + 14 * q^83 - 21 * q^85 + q^86 + 5 * q^88 - 8 * q^89 + 2 * q^91 - 3 * q^92 - 16 * q^94 + 23 * q^95 + 8 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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