LMFDB - Embedded newform 3726.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3726,2,Mod(1,3726)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3726, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3726.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3726 = 2 \cdot 3^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3726.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:	$29.7522597931$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	3726.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.44949 q^{5} +2.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.44949 q^{5} +2.00000 q^{7} +1.00000 q^{8} +1.44949 q^{10} +3.44949 q^{11} -6.89898 q^{13} +2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -2.00000 q^{19} +1.44949 q^{20} +3.44949 q^{22} -1.00000 q^{23} -2.89898 q^{25} -6.89898 q^{26} +2.00000 q^{28} +8.89898 q^{29} +7.89898 q^{31} +1.00000 q^{32} +2.00000 q^{34} +2.89898 q^{35} +7.44949 q^{37} -2.00000 q^{38} +1.44949 q^{40} +7.89898 q^{41} +10.0000 q^{43} +3.44949 q^{44} -1.00000 q^{46} -9.79796 q^{47} -3.00000 q^{49} -2.89898 q^{50} -6.89898 q^{52} -2.55051 q^{53} +5.00000 q^{55} +2.00000 q^{56} +8.89898 q^{58} +6.00000 q^{59} +10.5505 q^{61} +7.89898 q^{62} +1.00000 q^{64} -10.0000 q^{65} -8.34847 q^{67} +2.00000 q^{68} +2.89898 q^{70} -9.00000 q^{71} +15.7980 q^{73} +7.44949 q^{74} -2.00000 q^{76} +6.89898 q^{77} +10.8990 q^{79} +1.44949 q^{80} +7.89898 q^{82} -17.2474 q^{83} +2.89898 q^{85} +10.0000 q^{86} +3.44949 q^{88} -13.7980 q^{91} -1.00000 q^{92} -9.79796 q^{94} -2.89898 q^{95} -9.79796 q^{97} -3.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 2 q^{8} - 2 q^{10} + 2 q^{11} - 4 q^{13} + 4 q^{14} + 2 q^{16} + 4 q^{17} - 4 q^{19} - 2 q^{20} + 2 q^{22} - 2 q^{23} + 4 q^{25} - 4 q^{26} + 4 q^{28} + 8 q^{29} + 6 q^{31} + 2 q^{32} + 4 q^{34} - 4 q^{35} + 10 q^{37} - 4 q^{38} - 2 q^{40} + 6 q^{41} + 20 q^{43} + 2 q^{44} - 2 q^{46} - 6 q^{49} + 4 q^{50} - 4 q^{52} - 10 q^{53} + 10 q^{55} + 4 q^{56} + 8 q^{58} + 12 q^{59} + 26 q^{61} + 6 q^{62} + 2 q^{64} - 20 q^{65} - 2 q^{67} + 4 q^{68} - 4 q^{70} - 18 q^{71} + 12 q^{73} + 10 q^{74} - 4 q^{76} + 4 q^{77} + 12 q^{79} - 2 q^{80} + 6 q^{82} - 10 q^{83} - 4 q^{85} + 20 q^{86} + 2 q^{88} - 8 q^{91} - 2 q^{92} + 4 q^{95} - 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 + 2 * q^8 - 2 * q^10 + 2 * q^11 - 4 * q^13 + 4 * q^14 + 2 * q^16 + 4 * q^17 - 4 * q^19 - 2 * q^20 + 2 * q^22 - 2 * q^23 + 4 * q^25 - 4 * q^26 + 4 * q^28 + 8 * q^29 + 6 * q^31 + 2 * q^32 + 4 * q^34 - 4 * q^35 + 10 * q^37 - 4 * q^38 - 2 * q^40 + 6 * q^41 + 20 * q^43 + 2 * q^44 - 2 * q^46 - 6 * q^49 + 4 * q^50 - 4 * q^52 - 10 * q^53 + 10 * q^55 + 4 * q^56 + 8 * q^58 + 12 * q^59 + 26 * q^61 + 6 * q^62 + 2 * q^64 - 20 * q^65 - 2 * q^67 + 4 * q^68 - 4 * q^70 - 18 * q^71 + 12 * q^73 + 10 * q^74 - 4 * q^76 + 4 * q^77 + 12 * q^79 - 2 * q^80 + 6 * q^82 - 10 * q^83 - 4 * q^85 + 20 * q^86 + 2 * q^88 - 8 * q^91 - 2 * q^92 + 4 * q^95 - 6 * 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
$5$	1.44949	0.648232	0.324116	−	0.946017i	$-0.394933\pi$
$5$	1.44949	0.648232	0.324116	+	0.946017i	$0.394933\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.44949	0.458369
$11$	3.44949	1.04006	0.520030	−	0.854148i	$-0.325921\pi$
$11$	3.44949	1.04006	0.520030	+	0.854148i	$0.325921\pi$
$12$	0	0
$13$	−6.89898	−1.91343	−0.956716	−	0.291022i	$-0.906005\pi$
$13$	−6.89898	−1.91343	−0.956716	+	0.291022i	$0.906005\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	1.44949	0.324116
$21$	0	0
$22$	3.44949	0.735434
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−2.89898	−0.579796
$26$	−6.89898	−1.35300
$27$	0	0
$28$	2.00000	0.377964
$29$	8.89898	1.65250	0.826250	−	0.563304i	$-0.190470\pi$
$29$	8.89898	1.65250	0.826250	+	0.563304i	$0.190470\pi$
$30$	0	0
$31$	7.89898	1.41870	0.709349	−	0.704857i	$-0.248989\pi$
$31$	7.89898	1.41870	0.709349	+	0.704857i	$0.248989\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	2.89898	0.490017
$36$	0	0
$37$	7.44949	1.22469	0.612344	−	0.790591i	$-0.290227\pi$
$37$	7.44949	1.22469	0.612344	+	0.790591i	$0.290227\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	1.44949	0.229184
$41$	7.89898	1.23361	0.616807	−	0.787115i	$-0.288426\pi$
$41$	7.89898	1.23361	0.616807	+	0.787115i	$0.288426\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	3.44949	0.520030
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−9.79796	−1.42918	−0.714590	−	0.699544i	$-0.753387\pi$
$47$	−9.79796	−1.42918	−0.714590	+	0.699544i	$0.753387\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−2.89898	−0.409978
$51$	0	0
$52$	−6.89898	−0.956716
$53$	−2.55051	−0.350340	−0.175170	−	0.984538i	$-0.556047\pi$
$53$	−2.55051	−0.350340	−0.175170	+	0.984538i	$0.556047\pi$
$54$	0	0
$55$	5.00000	0.674200
$56$	2.00000	0.267261
$57$	0	0
$58$	8.89898	1.16849
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	10.5505	1.35085	0.675427	−	0.737427i	$-0.263959\pi$
$61$	10.5505	1.35085	0.675427	+	0.737427i	$0.263959\pi$
$62$	7.89898	1.00317
$63$	0	0
$64$	1.00000	0.125000
$65$	−10.0000	−1.24035
$66$	0	0
$67$	−8.34847	−1.01993	−0.509964	−	0.860196i	$-0.670341\pi$
$67$	−8.34847	−1.01993	−0.509964	+	0.860196i	$0.670341\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	2.89898	0.346494
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	0	0
$73$	15.7980	1.84901	0.924506	−	0.381168i	$-0.124478\pi$
$73$	15.7980	1.84901	0.924506	+	0.381168i	$0.124478\pi$
$74$	7.44949	0.865986
$75$	0	0
$76$	−2.00000	−0.229416
$77$	6.89898	0.786212
$78$	0	0
$79$	10.8990	1.22623	0.613115	−	0.789993i	$-0.289916\pi$
$79$	10.8990	1.22623	0.613115	+	0.789993i	$0.289916\pi$
$80$	1.44949	0.162058
$81$	0	0
$82$	7.89898	0.872296
$83$	−17.2474	−1.89315	−0.946577	−	0.322479i	$-0.895484\pi$
$83$	−17.2474	−1.89315	−0.946577	+	0.322479i	$0.895484\pi$
$84$	0	0
$85$	2.89898	0.314438
$86$	10.0000	1.07833
$87$	0	0
$88$	3.44949	0.367717
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−13.7980	−1.44642
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−9.79796	−1.01058
$95$	−2.89898	−0.297429
$96$	0	0
$97$	−9.79796	−0.994832	−0.497416	−	0.867512i	$-0.665718\pi$
$97$	−9.79796	−0.994832	−0.497416	+	0.867512i	$0.665718\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	−2.89898	−0.289898
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	−0.898979	−0.0885791	−0.0442895	−	0.999019i	$-0.514102\pi$
$103$	−0.898979	−0.0885791	−0.0442895	+	0.999019i	$0.514102\pi$
$104$	−6.89898	−0.676501
$105$	0	0
$106$	−2.55051	−0.247727
$107$	−11.7980	−1.14055	−0.570276	−	0.821453i	$-0.693164\pi$
$107$	−11.7980	−1.14055	−0.570276	+	0.821453i	$0.693164\pi$
$108$	0	0
$109$	−14.6969	−1.40771	−0.703856	−	0.710343i	$-0.748540\pi$
$109$	−14.6969	−1.40771	−0.703856	+	0.710343i	$0.748540\pi$
$110$	5.00000	0.476731
$111$	0	0
$112$	2.00000	0.188982
$113$	−1.79796	−0.169138	−0.0845689	−	0.996418i	$-0.526951\pi$
$113$	−1.79796	−0.169138	−0.0845689	+	0.996418i	$0.526951\pi$
$114$	0	0
$115$	−1.44949	−0.135166
$116$	8.89898	0.826250
$117$	0	0
$118$	6.00000	0.552345
$119$	4.00000	0.366679
$120$	0	0
$121$	0.898979	0.0817254
$122$	10.5505	0.955198
$123$	0	0
$124$	7.89898	0.709349
$125$	−11.4495	−1.02407
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−10.0000	−0.877058
$131$	9.79796	0.856052	0.428026	−	0.903767i	$-0.359209\pi$
$131$	9.79796	0.856052	0.428026	+	0.903767i	$0.359209\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	−8.34847	−0.721198
$135$	0	0
$136$	2.00000	0.171499
$137$	15.7980	1.34971	0.674855	−	0.737950i	$-0.264206\pi$
$137$	15.7980	1.34971	0.674855	+	0.737950i	$0.264206\pi$
$138$	0	0
$139$	22.6969	1.92513	0.962565	−	0.271052i	$-0.0873717\pi$
$139$	22.6969	1.92513	0.962565	+	0.271052i	$0.0873717\pi$
$140$	2.89898	0.245008
$141$	0	0
$142$	−9.00000	−0.755263
$143$	−23.7980	−1.99009
$144$	0	0
$145$	12.8990	1.07120
$146$	15.7980	1.30745
$147$	0	0
$148$	7.44949	0.612344
$149$	−10.6969	−0.876327	−0.438164	−	0.898895i	$-0.644371\pi$
$149$	−10.6969	−0.876327	−0.438164	+	0.898895i	$0.644371\pi$
$150$	0	0
$151$	5.89898	0.480052	0.240026	−	0.970766i	$-0.422844\pi$
$151$	5.89898	0.480052	0.240026	+	0.970766i	$0.422844\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	6.89898	0.555936
$155$	11.4495	0.919645
$156$	0	0
$157$	−4.89898	−0.390981	−0.195491	−	0.980706i	$-0.562630\pi$
$157$	−4.89898	−0.390981	−0.195491	+	0.980706i	$0.562630\pi$
$158$	10.8990	0.867076
$159$	0	0
$160$	1.44949	0.114592
$161$	−2.00000	−0.157622
$162$	0	0
$163$	−10.6969	−0.837849	−0.418924	−	0.908021i	$-0.637593\pi$
$163$	−10.6969	−0.837849	−0.418924	+	0.908021i	$0.637593\pi$
$164$	7.89898	0.616807
$165$	0	0
$166$	−17.2474	−1.33866
$167$	17.0000	1.31550	0.657750	−	0.753237i	$-0.271508\pi$
$167$	17.0000	1.31550	0.657750	+	0.753237i	$0.271508\pi$
$168$	0	0
$169$	34.5959	2.66122
$170$	2.89898	0.222342
$171$	0	0
$172$	10.0000	0.762493
$173$	14.6969	1.11739	0.558694	−	0.829374i	$-0.311303\pi$
$173$	14.6969	1.11739	0.558694	+	0.829374i	$0.311303\pi$
$174$	0	0
$175$	−5.79796	−0.438285
$176$	3.44949	0.260015
$177$	0	0
$178$	0	0
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	−15.2474	−1.13333	−0.566667	−	0.823947i	$-0.691767\pi$
$181$	−15.2474	−1.13333	−0.566667	+	0.823947i	$0.691767\pi$
$182$	−13.7980	−1.02277
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	10.7980	0.793882
$186$	0	0
$187$	6.89898	0.504503
$188$	−9.79796	−0.714590
$189$	0	0
$190$	−2.89898	−0.210314
$191$	−21.5959	−1.56263	−0.781313	−	0.624140i	$-0.785450\pi$
$191$	−21.5959	−1.56263	−0.781313	+	0.624140i	$0.785450\pi$
$192$	0	0
$193$	−10.7980	−0.777254	−0.388627	−	0.921395i	$-0.627051\pi$
$193$	−10.7980	−0.777254	−0.388627	+	0.921395i	$0.627051\pi$
$194$	−9.79796	−0.703452
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−21.7980	−1.55304	−0.776520	−	0.630092i	$-0.783017\pi$
$197$	−21.7980	−1.55304	−0.776520	+	0.630092i	$0.783017\pi$
$198$	0	0
$199$	21.5959	1.53089	0.765447	−	0.643499i	$-0.222518\pi$
$199$	21.5959	1.53089	0.765447	+	0.643499i	$0.222518\pi$
$200$	−2.89898	−0.204989
$201$	0	0
$202$	4.00000	0.281439
$203$	17.7980	1.24917
$204$	0	0
$205$	11.4495	0.799667
$206$	−0.898979	−0.0626349
$207$	0	0
$208$	−6.89898	−0.478358
$209$	−6.89898	−0.477212
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	−2.55051	−0.175170
$213$	0	0
$214$	−11.7980	−0.806492
$215$	14.4949	0.988544
$216$	0	0
$217$	15.7980	1.07244
$218$	−14.6969	−0.995402
$219$	0	0
$220$	5.00000	0.337100
$221$	−13.7980	−0.928151
$222$	0	0
$223$	−2.10102	−0.140695	−0.0703474	−	0.997523i	$-0.522411\pi$
$223$	−2.10102	−0.140695	−0.0703474	+	0.997523i	$0.522411\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−1.79796	−0.119598
$227$	27.7980	1.84502	0.922508	−	0.385979i	$-0.126136\pi$
$227$	27.7980	1.84502	0.922508	+	0.385979i	$0.126136\pi$
$228$	0	0
$229$	22.6969	1.49986	0.749928	−	0.661520i	$-0.230088\pi$
$229$	22.6969	1.49986	0.749928	+	0.661520i	$0.230088\pi$
$230$	−1.44949	−0.0955765
$231$	0	0
$232$	8.89898	0.584247
$233$	−13.5959	−0.890698	−0.445349	−	0.895357i	$-0.646920\pi$
$233$	−13.5959	−0.890698	−0.445349	+	0.895357i	$0.646920\pi$
$234$	0	0
$235$	−14.2020	−0.926439
$236$	6.00000	0.390567
$237$	0	0
$238$	4.00000	0.259281
$239$	1.89898	0.122835	0.0614174	−	0.998112i	$-0.480438\pi$
$239$	1.89898	0.122835	0.0614174	+	0.998112i	$0.480438\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0.898979	0.0577886
$243$	0	0
$244$	10.5505	0.675427
$245$	−4.34847	−0.277814
$246$	0	0
$247$	13.7980	0.877943
$248$	7.89898	0.501586
$249$	0	0
$250$	−11.4495	−0.724129
$251$	−19.4495	−1.22764	−0.613820	−	0.789446i	$-0.710368\pi$
$251$	−19.4495	−1.22764	−0.613820	+	0.789446i	$0.710368\pi$
$252$	0	0
$253$	−3.44949	−0.216868
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−11.8990	−0.742238	−0.371119	−	0.928585i	$-0.621026\pi$
$257$	−11.8990	−0.742238	−0.371119	+	0.928585i	$0.621026\pi$
$258$	0	0
$259$	14.8990	0.925778
$260$	−10.0000	−0.620174
$261$	0	0
$262$	9.79796	0.605320
$263$	−18.8990	−1.16536	−0.582680	−	0.812701i	$-0.697996\pi$
$263$	−18.8990	−1.16536	−0.582680	+	0.812701i	$0.697996\pi$
$264$	0	0
$265$	−3.69694	−0.227101
$266$	−4.00000	−0.245256
$267$	0	0
$268$	−8.34847	−0.509964
$269$	−8.89898	−0.542580	−0.271290	−	0.962498i	$-0.587450\pi$
$269$	−8.89898	−0.542580	−0.271290	+	0.962498i	$0.587450\pi$
$270$	0	0
$271$	8.79796	0.534438	0.267219	−	0.963636i	$-0.413895\pi$
$271$	8.79796	0.534438	0.267219	+	0.963636i	$0.413895\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	15.7980	0.954390
$275$	−10.0000	−0.603023
$276$	0	0
$277$	6.20204	0.372645	0.186322	−	0.982489i	$-0.440343\pi$
$277$	6.20204	0.372645	0.186322	+	0.982489i	$0.440343\pi$
$278$	22.6969	1.36127
$279$	0	0
$280$	2.89898	0.173247
$281$	−7.10102	−0.423611	−0.211806	−	0.977312i	$-0.567934\pi$
$281$	−7.10102	−0.423611	−0.211806	+	0.977312i	$0.567934\pi$
$282$	0	0
$283$	10.5505	0.627163	0.313581	−	0.949561i	$-0.398471\pi$
$283$	10.5505	0.627163	0.313581	+	0.949561i	$0.398471\pi$
$284$	−9.00000	−0.534052
$285$	0	0
$286$	−23.7980	−1.40720
$287$	15.7980	0.932524
$288$	0	0
$289$	−13.0000	−0.764706
$290$	12.8990	0.757454
$291$	0	0
$292$	15.7980	0.924506
$293$	1.44949	0.0846801	0.0423400	−	0.999103i	$-0.486519\pi$
$293$	1.44949	0.0846801	0.0423400	+	0.999103i	$0.486519\pi$
$294$	0	0
$295$	8.69694	0.506355
$296$	7.44949	0.432993
$297$	0	0
$298$	−10.6969	−0.619657
$299$	6.89898	0.398978
$300$	0	0
$301$	20.0000	1.15278
$302$	5.89898	0.339448
$303$	0	0
$304$	−2.00000	−0.114708
$305$	15.2929	0.875666
$306$	0	0
$307$	−14.8990	−0.850330	−0.425165	−	0.905116i	$-0.639784\pi$
$307$	−14.8990	−0.850330	−0.425165	+	0.905116i	$0.639784\pi$
$308$	6.89898	0.393106
$309$	0	0
$310$	11.4495	0.650287
$311$	−0.101021	−0.00572835	−0.00286417	−	0.999996i	$-0.500912\pi$
$311$	−0.101021	−0.00572835	−0.00286417	+	0.999996i	$0.500912\pi$
$312$	0	0
$313$	−16.6969	−0.943767	−0.471883	−	0.881661i	$-0.656426\pi$
$313$	−16.6969	−0.943767	−0.471883	+	0.881661i	$0.656426\pi$
$314$	−4.89898	−0.276465
$315$	0	0
$316$	10.8990	0.613115
$317$	14.8990	0.836810	0.418405	−	0.908261i	$-0.362589\pi$
$317$	14.8990	0.836810	0.418405	+	0.908261i	$0.362589\pi$
$318$	0	0
$319$	30.6969	1.71870
$320$	1.44949	0.0810289
$321$	0	0
$322$	−2.00000	−0.111456
$323$	−4.00000	−0.222566
$324$	0	0
$325$	20.0000	1.10940
$326$	−10.6969	−0.592449
$327$	0	0
$328$	7.89898	0.436148
$329$	−19.5959	−1.08036
$330$	0	0
$331$	−8.89898	−0.489132	−0.244566	−	0.969633i	$-0.578646\pi$
$331$	−8.89898	−0.489132	−0.244566	+	0.969633i	$0.578646\pi$
$332$	−17.2474	−0.946577
$333$	0	0
$334$	17.0000	0.930199
$335$	−12.1010	−0.661149
$336$	0	0
$337$	−10.2020	−0.555741	−0.277870	−	0.960619i	$-0.589629\pi$
$337$	−10.2020	−0.555741	−0.277870	+	0.960619i	$0.589629\pi$
$338$	34.5959	1.88177
$339$	0	0
$340$	2.89898	0.157219
$341$	27.2474	1.47553
$342$	0	0
$343$	−20.0000	−1.07990
$344$	10.0000	0.539164
$345$	0	0
$346$	14.6969	0.790112
$347$	−20.4949	−1.10022	−0.550112	−	0.835091i	$-0.685415\pi$
$347$	−20.4949	−1.10022	−0.550112	+	0.835091i	$0.685415\pi$
$348$	0	0
$349$	12.8990	0.690467	0.345233	−	0.938517i	$-0.387800\pi$
$349$	12.8990	0.690467	0.345233	+	0.938517i	$0.387800\pi$
$350$	−5.79796	−0.309914
$351$	0	0
$352$	3.44949	0.183858
$353$	−23.0000	−1.22417	−0.612083	−	0.790793i	$-0.709668\pi$
$353$	−23.0000	−1.22417	−0.612083	+	0.790793i	$0.709668\pi$
$354$	0	0
$355$	−13.0454	−0.692378
$356$	0	0
$357$	0	0
$358$	−2.00000	−0.105703
$359$	18.4949	0.976123	0.488062	−	0.872809i	$-0.337704\pi$
$359$	18.4949	0.976123	0.488062	+	0.872809i	$0.337704\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−15.2474	−0.801388
$363$	0	0
$364$	−13.7980	−0.723210
$365$	22.8990	1.19859
$366$	0	0
$367$	3.30306	0.172418	0.0862092	−	0.996277i	$-0.472525\pi$
$367$	3.30306	0.172418	0.0862092	+	0.996277i	$0.472525\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	10.7980	0.561359
$371$	−5.10102	−0.264832
$372$	0	0
$373$	−16.5505	−0.856953	−0.428476	−	0.903553i	$-0.640950\pi$
$373$	−16.5505	−0.856953	−0.428476	+	0.903553i	$0.640950\pi$
$374$	6.89898	0.356738
$375$	0	0
$376$	−9.79796	−0.505291
$377$	−61.3939	−3.16195
$378$	0	0
$379$	24.1464	1.24032	0.620159	−	0.784476i	$-0.287068\pi$
$379$	24.1464	1.24032	0.620159	+	0.784476i	$0.287068\pi$
$380$	−2.89898	−0.148715
$381$	0	0
$382$	−21.5959	−1.10494
$383$	−21.7980	−1.11382	−0.556912	−	0.830572i	$-0.688014\pi$
$383$	−21.7980	−1.11382	−0.556912	+	0.830572i	$0.688014\pi$
$384$	0	0
$385$	10.0000	0.509647
$386$	−10.7980	−0.549602
$387$	0	0
$388$	−9.79796	−0.497416
$389$	17.9444	0.909817	0.454908	−	0.890538i	$-0.349672\pi$
$389$	17.9444	0.909817	0.454908	+	0.890538i	$0.349672\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	−3.00000	−0.151523
$393$	0	0
$394$	−21.7980	−1.09817
$395$	15.7980	0.794882
$396$	0	0
$397$	−16.4949	−0.827855	−0.413928	−	0.910310i	$-0.635843\pi$
$397$	−16.4949	−0.827855	−0.413928	+	0.910310i	$0.635843\pi$
$398$	21.5959	1.08251
$399$	0	0
$400$	−2.89898	−0.144949
$401$	−25.1010	−1.25349	−0.626743	−	0.779226i	$-0.715612\pi$
$401$	−25.1010	−1.25349	−0.626743	+	0.779226i	$0.715612\pi$
$402$	0	0
$403$	−54.4949	−2.71458
$404$	4.00000	0.199007
$405$	0	0
$406$	17.7980	0.883298
$407$	25.6969	1.27375
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	11.4495	0.565450
$411$	0	0
$412$	−0.898979	−0.0442895
$413$	12.0000	0.590481
$414$	0	0
$415$	−25.0000	−1.22720
$416$	−6.89898	−0.338250
$417$	0	0
$418$	−6.89898	−0.337440
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	−37.9444	−1.84930	−0.924648	−	0.380823i	$-0.875641\pi$
$421$	−37.9444	−1.84930	−0.924648	+	0.380823i	$0.875641\pi$
$422$	−14.0000	−0.681509
$423$	0	0
$424$	−2.55051	−0.123864
$425$	−5.79796	−0.281242
$426$	0	0
$427$	21.1010	1.02115
$428$	−11.7980	−0.570276
$429$	0	0
$430$	14.4949	0.699006
$431$	−2.69694	−0.129907	−0.0649535	−	0.997888i	$-0.520690\pi$
$431$	−2.69694	−0.129907	−0.0649535	+	0.997888i	$0.520690\pi$
$432$	0	0
$433$	24.6969	1.18686	0.593430	−	0.804886i	$-0.297774\pi$
$433$	24.6969	1.18686	0.593430	+	0.804886i	$0.297774\pi$
$434$	15.7980	0.758326
$435$	0	0
$436$	−14.6969	−0.703856
$437$	2.00000	0.0956730
$438$	0	0
$439$	7.20204	0.343735	0.171867	−	0.985120i	$-0.445020\pi$
$439$	7.20204	0.343735	0.171867	+	0.985120i	$0.445020\pi$
$440$	5.00000	0.238366
$441$	0	0
$442$	−13.7980	−0.656302
$443$	−21.5959	−1.02605	−0.513027	−	0.858373i	$-0.671476\pi$
$443$	−21.5959	−1.02605	−0.513027	+	0.858373i	$0.671476\pi$
$444$	0	0
$445$	0	0
$446$	−2.10102	−0.0994862
$447$	0	0
$448$	2.00000	0.0944911
$449$	−19.6969	−0.929556	−0.464778	−	0.885427i	$-0.653866\pi$
$449$	−19.6969	−0.929556	−0.464778	+	0.885427i	$0.653866\pi$
$450$	0	0
$451$	27.2474	1.28303
$452$	−1.79796	−0.0845689
$453$	0	0
$454$	27.7980	1.30462
$455$	−20.0000	−0.937614
$456$	0	0
$457$	−11.1010	−0.519284	−0.259642	−	0.965705i	$-0.583605\pi$
$457$	−11.1010	−0.519284	−0.259642	+	0.965705i	$0.583605\pi$
$458$	22.6969	1.06056
$459$	0	0
$460$	−1.44949	−0.0675828
$461$	−14.2020	−0.661455	−0.330727	−	0.943726i	$-0.607294\pi$
$461$	−14.2020	−0.661455	−0.330727	+	0.943726i	$0.607294\pi$
$462$	0	0
$463$	20.5959	0.957173	0.478587	−	0.878040i	$-0.341149\pi$
$463$	20.5959	0.957173	0.478587	+	0.878040i	$0.341149\pi$
$464$	8.89898	0.413125
$465$	0	0
$466$	−13.5959	−0.629819
$467$	11.2474	0.520470	0.260235	−	0.965545i	$-0.416200\pi$
$467$	11.2474	0.520470	0.260235	+	0.965545i	$0.416200\pi$
$468$	0	0
$469$	−16.6969	−0.770993
$470$	−14.2020	−0.655091
$471$	0	0
$472$	6.00000	0.276172
$473$	34.4949	1.58608
$474$	0	0
$475$	5.79796	0.266029
$476$	4.00000	0.183340
$477$	0	0
$478$	1.89898	0.0868573
$479$	5.79796	0.264916	0.132458	−	0.991189i	$-0.457713\pi$
$479$	5.79796	0.264916	0.132458	+	0.991189i	$0.457713\pi$
$480$	0	0
$481$	−51.3939	−2.34336
$482$	10.0000	0.455488
$483$	0	0
$484$	0.898979	0.0408627
$485$	−14.2020	−0.644881
$486$	0	0
$487$	14.5959	0.661404	0.330702	−	0.943735i	$-0.392714\pi$
$487$	14.5959	0.661404	0.330702	+	0.943735i	$0.392714\pi$
$488$	10.5505	0.477599
$489$	0	0
$490$	−4.34847	−0.196444
$491$	−7.10102	−0.320465	−0.160232	−	0.987079i	$-0.551224\pi$
$491$	−7.10102	−0.320465	−0.160232	+	0.987079i	$0.551224\pi$
$492$	0	0
$493$	17.7980	0.801580
$494$	13.7980	0.620800
$495$	0	0
$496$	7.89898	0.354675
$497$	−18.0000	−0.807410
$498$	0	0
$499$	33.3939	1.49492	0.747458	−	0.664309i	$-0.231274\pi$
$499$	33.3939	1.49492	0.747458	+	0.664309i	$0.231274\pi$
$500$	−11.4495	−0.512037
$501$	0	0
$502$	−19.4495	−0.868073
$503$	10.8990	0.485961	0.242981	−	0.970031i	$-0.421875\pi$
$503$	10.8990	0.485961	0.242981	+	0.970031i	$0.421875\pi$
$504$	0	0
$505$	5.79796	0.258006
$506$	−3.44949	−0.153349
$507$	0	0
$508$	−8.00000	−0.354943
$509$	15.7980	0.700232	0.350116	−	0.936706i	$-0.386142\pi$
$509$	15.7980	0.700232	0.350116	+	0.936706i	$0.386142\pi$
$510$	0	0
$511$	31.5959	1.39772
$512$	1.00000	0.0441942
$513$	0	0
$514$	−11.8990	−0.524841
$515$	−1.30306	−0.0574198
$516$	0	0
$517$	−33.7980	−1.48643
$518$	14.8990	0.654624
$519$	0	0
$520$	−10.0000	−0.438529
$521$	−35.1918	−1.54178	−0.770891	−	0.636967i	$-0.780189\pi$
$521$	−35.1918	−1.54178	−0.770891	+	0.636967i	$0.780189\pi$
$522$	0	0
$523$	6.34847	0.277599	0.138800	−	0.990320i	$-0.455676\pi$
$523$	6.34847	0.277599	0.138800	+	0.990320i	$0.455676\pi$
$524$	9.79796	0.428026
$525$	0	0
$526$	−18.8990	−0.824035
$527$	15.7980	0.688170
$528$	0	0
$529$	1.00000	0.0434783
$530$	−3.69694	−0.160585
$531$	0	0
$532$	−4.00000	−0.173422
$533$	−54.4949	−2.36044
$534$	0	0
$535$	−17.1010	−0.739342
$536$	−8.34847	−0.360599
$537$	0	0
$538$	−8.89898	−0.383662
$539$	−10.3485	−0.445740
$540$	0	0
$541$	−19.5959	−0.842494	−0.421247	−	0.906946i	$-0.638408\pi$
$541$	−19.5959	−0.842494	−0.421247	+	0.906946i	$0.638408\pi$
$542$	8.79796	0.377905
$543$	0	0
$544$	2.00000	0.0857493
$545$	−21.3031	−0.912523
$546$	0	0
$547$	6.49490	0.277702	0.138851	−	0.990313i	$-0.455659\pi$
$547$	6.49490	0.277702	0.138851	+	0.990313i	$0.455659\pi$
$548$	15.7980	0.674855
$549$	0	0
$550$	−10.0000	−0.426401
$551$	−17.7980	−0.758219
$552$	0	0
$553$	21.7980	0.926944
$554$	6.20204	0.263499
$555$	0	0
$556$	22.6969	0.962565
$557$	37.9444	1.60776	0.803878	−	0.594795i	$-0.202767\pi$
$557$	37.9444	1.60776	0.803878	+	0.594795i	$0.202767\pi$
$558$	0	0
$559$	−68.9898	−2.91796
$560$	2.89898	0.122504
$561$	0	0
$562$	−7.10102	−0.299538
$563$	26.0000	1.09577	0.547885	−	0.836554i	$-0.315433\pi$
$563$	26.0000	1.09577	0.547885	+	0.836554i	$0.315433\pi$
$564$	0	0
$565$	−2.60612	−0.109640
$566$	10.5505	0.443471
$567$	0	0
$568$	−9.00000	−0.377632
$569$	−33.1010	−1.38767	−0.693833	−	0.720135i	$-0.744080\pi$
$569$	−33.1010	−1.38767	−0.693833	+	0.720135i	$0.744080\pi$
$570$	0	0
$571$	11.2474	0.470691	0.235346	−	0.971912i	$-0.424378\pi$
$571$	11.2474	0.470691	0.235346	+	0.971912i	$0.424378\pi$
$572$	−23.7980	−0.995043
$573$	0	0
$574$	15.7980	0.659394
$575$	2.89898	0.120896
$576$	0	0
$577$	17.6969	0.736733	0.368367	−	0.929681i	$-0.379917\pi$
$577$	17.6969	0.736733	0.368367	+	0.929681i	$0.379917\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	12.8990	0.535601
$581$	−34.4949	−1.43109
$582$	0	0
$583$	−8.79796	−0.364374
$584$	15.7980	0.653724
$585$	0	0
$586$	1.44949	0.0598779
$587$	35.1010	1.44877	0.724387	−	0.689393i	$-0.242123\pi$
$587$	35.1010	1.44877	0.724387	+	0.689393i	$0.242123\pi$
$588$	0	0
$589$	−15.7980	−0.650944
$590$	8.69694	0.358047
$591$	0	0
$592$	7.44949	0.306172
$593$	−15.4949	−0.636299	−0.318150	−	0.948041i	$-0.603061\pi$
$593$	−15.4949	−0.636299	−0.318150	+	0.948041i	$0.603061\pi$
$594$	0	0
$595$	5.79796	0.237693
$596$	−10.6969	−0.438164
$597$	0	0
$598$	6.89898	0.282120
$599$	−11.8990	−0.486179	−0.243090	−	0.970004i	$-0.578161\pi$
$599$	−11.8990	−0.486179	−0.243090	+	0.970004i	$0.578161\pi$
$600$	0	0
$601$	−21.8990	−0.893278	−0.446639	−	0.894714i	$-0.647379\pi$
$601$	−21.8990	−0.893278	−0.446639	+	0.894714i	$0.647379\pi$
$602$	20.0000	0.815139
$603$	0	0
$604$	5.89898	0.240026
$605$	1.30306	0.0529770
$606$	0	0
$607$	30.1010	1.22176	0.610881	−	0.791722i	$-0.290815\pi$
$607$	30.1010	1.22176	0.610881	+	0.791722i	$0.290815\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	15.2929	0.619190
$611$	67.5959	2.73464
$612$	0	0
$613$	−11.1010	−0.448366	−0.224183	−	0.974547i	$-0.571971\pi$
$613$	−11.1010	−0.448366	−0.224183	+	0.974547i	$0.571971\pi$
$614$	−14.8990	−0.601274
$615$	0	0
$616$	6.89898	0.277968
$617$	−20.2020	−0.813304	−0.406652	−	0.913583i	$-0.633304\pi$
$617$	−20.2020	−0.813304	−0.406652	+	0.913583i	$0.633304\pi$
$618$	0	0
$619$	2.95459	0.118755	0.0593775	−	0.998236i	$-0.481088\pi$
$619$	2.95459	0.118755	0.0593775	+	0.998236i	$0.481088\pi$
$620$	11.4495	0.459823
$621$	0	0
$622$	−0.101021	−0.00405055
$623$	0	0
$624$	0	0
$625$	−2.10102	−0.0840408
$626$	−16.6969	−0.667344
$627$	0	0
$628$	−4.89898	−0.195491
$629$	14.8990	0.594061
$630$	0	0
$631$	−5.10102	−0.203068	−0.101534	−	0.994832i	$-0.532375\pi$
$631$	−5.10102	−0.203068	−0.101534	+	0.994832i	$0.532375\pi$
$632$	10.8990	0.433538
$633$	0	0
$634$	14.8990	0.591714
$635$	−11.5959	−0.460170
$636$	0	0
$637$	20.6969	0.820043
$638$	30.6969	1.21530
$639$	0	0
$640$	1.44949	0.0572961
$641$	−5.79796	−0.229006	−0.114503	−	0.993423i	$-0.536527\pi$
$641$	−5.79796	−0.229006	−0.114503	+	0.993423i	$0.536527\pi$
$642$	0	0
$643$	1.24745	0.0491946	0.0245973	−	0.999697i	$-0.492170\pi$
$643$	1.24745	0.0491946	0.0245973	+	0.999697i	$0.492170\pi$
$644$	−2.00000	−0.0788110
$645$	0	0
$646$	−4.00000	−0.157378
$647$	−11.4949	−0.451911	−0.225956	−	0.974138i	$-0.572550\pi$
$647$	−11.4949	−0.451911	−0.225956	+	0.974138i	$0.572550\pi$
$648$	0	0
$649$	20.6969	0.812426
$650$	20.0000	0.784465
$651$	0	0
$652$	−10.6969	−0.418924
$653$	−38.6969	−1.51433	−0.757164	−	0.653225i	$-0.773416\pi$
$653$	−38.6969	−1.51433	−0.757164	+	0.653225i	$0.773416\pi$
$654$	0	0
$655$	14.2020	0.554920
$656$	7.89898	0.308403
$657$	0	0
$658$	−19.5959	−0.763928
$659$	−3.79796	−0.147947	−0.0739737	−	0.997260i	$-0.523568\pi$
$659$	−3.79796	−0.147947	−0.0739737	+	0.997260i	$0.523568\pi$
$660$	0	0
$661$	4.34847	0.169136	0.0845679	−	0.996418i	$-0.473049\pi$
$661$	4.34847	0.169136	0.0845679	+	0.996418i	$0.473049\pi$
$662$	−8.89898	−0.345869
$663$	0	0
$664$	−17.2474	−0.669331
$665$	−5.79796	−0.224835
$666$	0	0
$667$	−8.89898	−0.344570
$668$	17.0000	0.657750
$669$	0	0
$670$	−12.1010	−0.467503
$671$	36.3939	1.40497
$672$	0	0
$673$	−16.2020	−0.624543	−0.312271	−	0.949993i	$-0.601090\pi$
$673$	−16.2020	−0.624543	−0.312271	+	0.949993i	$0.601090\pi$
$674$	−10.2020	−0.392968
$675$	0	0
$676$	34.5959	1.33061
$677$	−17.9444	−0.689659	−0.344829	−	0.938665i	$-0.612063\pi$
$677$	−17.9444	−0.689659	−0.344829	+	0.938665i	$0.612063\pi$
$678$	0	0
$679$	−19.5959	−0.752022
$680$	2.89898	0.111171
$681$	0	0
$682$	27.2474	1.04336
$683$	−17.3939	−0.665558	−0.332779	−	0.943005i	$-0.607986\pi$
$683$	−17.3939	−0.665558	−0.332779	+	0.943005i	$0.607986\pi$
$684$	0	0
$685$	22.8990	0.874925
$686$	−20.0000	−0.763604
$687$	0	0
$688$	10.0000	0.381246
$689$	17.5959	0.670351
$690$	0	0
$691$	−1.30306	−0.0495708	−0.0247854	−	0.999693i	$-0.507890\pi$
$691$	−1.30306	−0.0495708	−0.0247854	+	0.999693i	$0.507890\pi$
$692$	14.6969	0.558694
$693$	0	0
$694$	−20.4949	−0.777976
$695$	32.8990	1.24793
$696$	0	0
$697$	15.7980	0.598390
$698$	12.8990	0.488234
$699$	0	0
$700$	−5.79796	−0.219142
$701$	29.9444	1.13098	0.565492	−	0.824754i	$-0.308686\pi$
$701$	29.9444	1.13098	0.565492	+	0.824754i	$0.308686\pi$
$702$	0	0
$703$	−14.8990	−0.561926
$704$	3.44949	0.130008
$705$	0	0
$706$	−23.0000	−0.865616
$707$	8.00000	0.300871
$708$	0	0
$709$	33.6515	1.26381	0.631905	−	0.775046i	$-0.282273\pi$
$709$	33.6515	1.26381	0.631905	+	0.775046i	$0.282273\pi$
$710$	−13.0454	−0.489585
$711$	0	0
$712$	0	0
$713$	−7.89898	−0.295819
$714$	0	0
$715$	−34.4949	−1.29004
$716$	−2.00000	−0.0747435
$717$	0	0
$718$	18.4949	0.690223
$719$	−3.49490	−0.130338	−0.0651688	−	0.997874i	$-0.520759\pi$
$719$	−3.49490	−0.130338	−0.0651688	+	0.997874i	$0.520759\pi$
$720$	0	0
$721$	−1.79796	−0.0669595
$722$	−15.0000	−0.558242
$723$	0	0
$724$	−15.2474	−0.566667
$725$	−25.7980	−0.958112
$726$	0	0
$727$	−21.7980	−0.808442	−0.404221	−	0.914661i	$-0.632457\pi$
$727$	−21.7980	−0.808442	−0.404221	+	0.914661i	$0.632457\pi$
$728$	−13.7980	−0.511386
$729$	0	0
$730$	22.8990	0.847529
$731$	20.0000	0.739727
$732$	0	0
$733$	−30.2929	−1.11889	−0.559446	−	0.828867i	$-0.688986\pi$
$733$	−30.2929	−1.11889	−0.559446	+	0.828867i	$0.688986\pi$
$734$	3.30306	0.121918
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−28.7980	−1.06079
$738$	0	0
$739$	−3.10102	−0.114073	−0.0570364	−	0.998372i	$-0.518165\pi$
$739$	−3.10102	−0.114073	−0.0570364	+	0.998372i	$0.518165\pi$
$740$	10.7980	0.396941
$741$	0	0
$742$	−5.10102	−0.187264
$743$	−23.3939	−0.858238	−0.429119	−	0.903248i	$-0.641176\pi$
$743$	−23.3939	−0.858238	−0.429119	+	0.903248i	$0.641176\pi$
$744$	0	0
$745$	−15.5051	−0.568063
$746$	−16.5505	−0.605957
$747$	0	0
$748$	6.89898	0.252252
$749$	−23.5959	−0.862176
$750$	0	0
$751$	−26.2020	−0.956126	−0.478063	−	0.878326i	$-0.658661\pi$
$751$	−26.2020	−0.956126	−0.478063	+	0.878326i	$0.658661\pi$
$752$	−9.79796	−0.357295
$753$	0	0
$754$	−61.3939	−2.23583
$755$	8.55051	0.311185
$756$	0	0
$757$	−50.6413	−1.84059	−0.920295	−	0.391225i	$-0.872051\pi$
$757$	−50.6413	−1.84059	−0.920295	+	0.391225i	$0.872051\pi$
$758$	24.1464	0.877038
$759$	0	0
$760$	−2.89898	−0.105157
$761$	−37.5959	−1.36285	−0.681425	−	0.731888i	$-0.738640\pi$
$761$	−37.5959	−1.36285	−0.681425	+	0.731888i	$0.738640\pi$
$762$	0	0
$763$	−29.3939	−1.06413
$764$	−21.5959	−0.781313
$765$	0	0
$766$	−21.7980	−0.787592
$767$	−41.3939	−1.49465
$768$	0	0
$769$	36.6969	1.32333	0.661663	−	0.749802i	$-0.269851\pi$
$769$	36.6969	1.32333	0.661663	+	0.749802i	$0.269851\pi$
$770$	10.0000	0.360375
$771$	0	0
$772$	−10.7980	−0.388627
$773$	0.146428	0.00526666	0.00263333	−	0.999997i	$-0.499162\pi$
$773$	0.146428	0.00526666	0.00263333	+	0.999997i	$0.499162\pi$
$774$	0	0
$775$	−22.8990	−0.822556
$776$	−9.79796	−0.351726
$777$	0	0
$778$	17.9444	0.643337
$779$	−15.7980	−0.566021
$780$	0	0
$781$	−31.0454	−1.11089
$782$	−2.00000	−0.0715199
$783$	0	0
$784$	−3.00000	−0.107143
$785$	−7.10102	−0.253446
$786$	0	0
$787$	−18.0000	−0.641631	−0.320815	−	0.947142i	$-0.603957\pi$
$787$	−18.0000	−0.641631	−0.320815	+	0.947142i	$0.603957\pi$
$788$	−21.7980	−0.776520
$789$	0	0
$790$	15.7980	0.562066
$791$	−3.59592	−0.127856
$792$	0	0
$793$	−72.7878	−2.58477
$794$	−16.4949	−0.585382
$795$	0	0
$796$	21.5959	0.765447
$797$	12.1464	0.430249	0.215124	−	0.976587i	$-0.430984\pi$
$797$	12.1464	0.430249	0.215124	+	0.976587i	$0.430984\pi$
$798$	0	0
$799$	−19.5959	−0.693254
$800$	−2.89898	−0.102494
$801$	0	0
$802$	−25.1010	−0.886348
$803$	54.4949	1.92308
$804$	0	0
$805$	−2.89898	−0.102176
$806$	−54.4949	−1.91950
$807$	0	0
$808$	4.00000	0.140720
$809$	−3.79796	−0.133529	−0.0667646	−	0.997769i	$-0.521268\pi$
$809$	−3.79796	−0.133529	−0.0667646	+	0.997769i	$0.521268\pi$
$810$	0	0
$811$	22.2020	0.779619	0.389810	−	0.920895i	$-0.372541\pi$
$811$	22.2020	0.779619	0.389810	+	0.920895i	$0.372541\pi$
$812$	17.7980	0.624586
$813$	0	0
$814$	25.6969	0.900677
$815$	−15.5051	−0.543120
$816$	0	0
$817$	−20.0000	−0.699711
$818$	−5.00000	−0.174821
$819$	0	0
$820$	11.4495	0.399834
$821$	44.6969	1.55993	0.779967	−	0.625821i	$-0.215236\pi$
$821$	44.6969	1.55993	0.779967	+	0.625821i	$0.215236\pi$
$822$	0	0
$823$	−27.4949	−0.958412	−0.479206	−	0.877702i	$-0.659075\pi$
$823$	−27.4949	−0.958412	−0.479206	+	0.877702i	$0.659075\pi$
$824$	−0.898979	−0.0313174
$825$	0	0
$826$	12.0000	0.417533
$827$	36.5505	1.27099	0.635493	−	0.772107i	$-0.280797\pi$
$827$	36.5505	1.27099	0.635493	+	0.772107i	$0.280797\pi$
$828$	0	0
$829$	20.6969	0.718834	0.359417	−	0.933177i	$-0.382976\pi$
$829$	20.6969	0.718834	0.359417	+	0.933177i	$0.382976\pi$
$830$	−25.0000	−0.867763
$831$	0	0
$832$	−6.89898	−0.239179
$833$	−6.00000	−0.207888
$834$	0	0
$835$	24.6413	0.852748
$836$	−6.89898	−0.238606
$837$	0	0
$838$	−6.00000	−0.207267
$839$	13.1010	0.452297	0.226149	−	0.974093i	$-0.427386\pi$
$839$	13.1010	0.452297	0.226149	+	0.974093i	$0.427386\pi$
$840$	0	0
$841$	50.1918	1.73075
$842$	−37.9444	−1.30765
$843$	0	0
$844$	−14.0000	−0.481900
$845$	50.1464	1.72509
$846$	0	0
$847$	1.79796	0.0617786
$848$	−2.55051	−0.0875849
$849$	0	0
$850$	−5.79796	−0.198868
$851$	−7.44949	−0.255365
$852$	0	0
$853$	28.0000	0.958702	0.479351	−	0.877623i	$-0.340872\pi$
$853$	28.0000	0.958702	0.479351	+	0.877623i	$0.340872\pi$
$854$	21.1010	0.722062
$855$	0	0
$856$	−11.7980	−0.403246
$857$	45.0000	1.53717	0.768585	−	0.639747i	$-0.220961\pi$
$857$	45.0000	1.53717	0.768585	+	0.639747i	$0.220961\pi$
$858$	0	0
$859$	−35.7980	−1.22141	−0.610705	−	0.791858i	$-0.709114\pi$
$859$	−35.7980	−1.22141	−0.610705	+	0.791858i	$0.709114\pi$
$860$	14.4949	0.494272
$861$	0	0
$862$	−2.69694	−0.0918581
$863$	−0.505103	−0.0171939	−0.00859695	−	0.999963i	$-0.502737\pi$
$863$	−0.505103	−0.0171939	−0.00859695	+	0.999963i	$0.502737\pi$
$864$	0	0
$865$	21.3031	0.724326
$866$	24.6969	0.839236
$867$	0	0
$868$	15.7980	0.536218
$869$	37.5959	1.27535
$870$	0	0
$871$	57.5959	1.95156
$872$	−14.6969	−0.497701
$873$	0	0
$874$	2.00000	0.0676510
$875$	−22.8990	−0.774127
$876$	0	0
$877$	−37.1010	−1.25281	−0.626406	−	0.779497i	$-0.715475\pi$
$877$	−37.1010	−1.25281	−0.626406	+	0.779497i	$0.715475\pi$
$878$	7.20204	0.243057
$879$	0	0
$880$	5.00000	0.168550
$881$	25.5959	0.862348	0.431174	−	0.902269i	$-0.358099\pi$
$881$	25.5959	0.862348	0.431174	+	0.902269i	$0.358099\pi$
$882$	0	0
$883$	−6.00000	−0.201916	−0.100958	−	0.994891i	$-0.532191\pi$
$883$	−6.00000	−0.201916	−0.100958	+	0.994891i	$0.532191\pi$
$884$	−13.7980	−0.464076
$885$	0	0
$886$	−21.5959	−0.725529
$887$	−6.79796	−0.228253	−0.114127	−	0.993466i	$-0.536407\pi$
$887$	−6.79796	−0.228253	−0.114127	+	0.993466i	$0.536407\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	−2.10102	−0.0703474
$893$	19.5959	0.655752
$894$	0	0
$895$	−2.89898	−0.0969022
$896$	2.00000	0.0668153
$897$	0	0
$898$	−19.6969	−0.657295
$899$	70.2929	2.34440
$900$	0	0
$901$	−5.10102	−0.169940
$902$	27.2474	0.907241
$903$	0	0
$904$	−1.79796	−0.0597992
$905$	−22.1010	−0.734663
$906$	0	0
$907$	−6.34847	−0.210797	−0.105399	−	0.994430i	$-0.533612\pi$
$907$	−6.34847	−0.210797	−0.105399	+	0.994430i	$0.533612\pi$
$908$	27.7980	0.922508
$909$	0	0
$910$	−20.0000	−0.662994
$911$	44.2929	1.46749	0.733744	−	0.679426i	$-0.237771\pi$
$911$	44.2929	1.46749	0.733744	+	0.679426i	$0.237771\pi$
$912$	0	0
$913$	−59.4949	−1.96899
$914$	−11.1010	−0.367189
$915$	0	0
$916$	22.6969	0.749928
$917$	19.5959	0.647114
$918$	0	0
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	−1.44949	−0.0477883
$921$	0	0
$922$	−14.2020	−0.467719
$923$	62.0908	2.04374
$924$	0	0
$925$	−21.5959	−0.710069
$926$	20.5959	0.676824
$927$	0	0
$928$	8.89898	0.292123
$929$	−35.6969	−1.17118	−0.585589	−	0.810608i	$-0.699137\pi$
$929$	−35.6969	−1.17118	−0.585589	+	0.810608i	$0.699137\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	−13.5959	−0.445349
$933$	0	0
$934$	11.2474	0.368028
$935$	10.0000	0.327035
$936$	0	0
$937$	54.0908	1.76707	0.883535	−	0.468365i	$-0.155157\pi$
$937$	54.0908	1.76707	0.883535	+	0.468365i	$0.155157\pi$
$938$	−16.6969	−0.545174
$939$	0	0
$940$	−14.2020	−0.463220
$941$	−35.9444	−1.17175	−0.585877	−	0.810400i	$-0.699250\pi$
$941$	−35.9444	−1.17175	−0.585877	+	0.810400i	$0.699250\pi$
$942$	0	0
$943$	−7.89898	−0.257226
$944$	6.00000	0.195283
$945$	0	0
$946$	34.4949	1.12153
$947$	52.2929	1.69929	0.849645	−	0.527355i	$-0.176816\pi$
$947$	52.2929	1.69929	0.849645	+	0.527355i	$0.176816\pi$
$948$	0	0
$949$	−108.990	−3.53796
$950$	5.79796	0.188111
$951$	0	0
$952$	4.00000	0.129641
$953$	34.6969	1.12394	0.561972	−	0.827156i	$-0.310043\pi$
$953$	34.6969	1.12394	0.561972	+	0.827156i	$0.310043\pi$
$954$	0	0
$955$	−31.3031	−1.01294
$956$	1.89898	0.0614174
$957$	0	0
$958$	5.79796	0.187324
$959$	31.5959	1.02029
$960$	0	0
$961$	31.3939	1.01271
$962$	−51.3939	−1.65701
$963$	0	0
$964$	10.0000	0.322078
$965$	−15.6515	−0.503841
$966$	0	0
$967$	−15.6969	−0.504780	−0.252390	−	0.967626i	$-0.581217\pi$
$967$	−15.6969	−0.504780	−0.252390	+	0.967626i	$0.581217\pi$
$968$	0.898979	0.0288943
$969$	0	0
$970$	−14.2020	−0.456000
$971$	−51.3939	−1.64931	−0.824654	−	0.565638i	$-0.808630\pi$
$971$	−51.3939	−1.64931	−0.824654	+	0.565638i	$0.808630\pi$
$972$	0	0
$973$	45.3939	1.45526
$974$	14.5959	0.467683
$975$	0	0
$976$	10.5505	0.337714
$977$	−24.4949	−0.783661	−0.391831	−	0.920037i	$-0.628158\pi$
$977$	−24.4949	−0.783661	−0.391831	+	0.920037i	$0.628158\pi$
$978$	0	0
$979$	0	0
$980$	−4.34847	−0.138907
$981$	0	0
$982$	−7.10102	−0.226603
$983$	3.79796	0.121136	0.0605680	−	0.998164i	$-0.480709\pi$
$983$	3.79796	0.121136	0.0605680	+	0.998164i	$0.480709\pi$
$984$	0	0
$985$	−31.5959	−1.00673
$986$	17.7980	0.566802
$987$	0	0
$988$	13.7980	0.438972
$989$	−10.0000	−0.317982
$990$	0	0
$991$	−13.3939	−0.425471	−0.212735	−	0.977110i	$-0.568237\pi$
$991$	−13.3939	−0.425471	−0.212735	+	0.977110i	$0.568237\pi$
$992$	7.89898	0.250793
$993$	0	0
$994$	−18.0000	−0.570925
$995$	31.3031	0.992374
$996$	0	0
$997$	−24.8990	−0.788559	−0.394279	−	0.918991i	$-0.629006\pi$
$997$	−24.8990	−0.788559	−0.394279	+	0.918991i	$0.629006\pi$
$998$	33.3939	1.05706
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3726.2.a.l.1.2	yes	2
3.2	odd	2		3726.2.a.j.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3726.2.a.j.1.1	✓	2	3.2	odd	2
3726.2.a.l.1.2	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3726 = 2 \cdot 3^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3726.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.44949 q^{5} +2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.44949 q^{5} +2.00000 q^{7} +1.00000 q^{8} +1.44949 q^{10} +3.44949 q^{11} -6.89898 q^{13} +2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -2.00000 q^{19} +1.44949 q^{20} +3.44949 q^{22} -1.00000 q^{23} -2.89898 q^{25} -6.89898 q^{26} +2.00000 q^{28} +8.89898 q^{29} +7.89898 q^{31} +1.00000 q^{32} +2.00000 q^{34} +2.89898 q^{35} +7.44949 q^{37} -2.00000 q^{38} +1.44949 q^{40} +7.89898 q^{41} +10.0000 q^{43} +3.44949 q^{44} -1.00000 q^{46} -9.79796 q^{47} -3.00000 q^{49} -2.89898 q^{50} -6.89898 q^{52} -2.55051 q^{53} +5.00000 q^{55} +2.00000 q^{56} +8.89898 q^{58} +6.00000 q^{59} +10.5505 q^{61} +7.89898 q^{62} +1.00000 q^{64} -10.0000 q^{65} -8.34847 q^{67} +2.00000 q^{68} +2.89898 q^{70} -9.00000 q^{71} +15.7980 q^{73} +7.44949 q^{74} -2.00000 q^{76} +6.89898 q^{77} +10.8990 q^{79} +1.44949 q^{80} +7.89898 q^{82} -17.2474 q^{83} +2.89898 q^{85} +10.0000 q^{86} +3.44949 q^{88} -13.7980 q^{91} -1.00000 q^{92} -9.79796 q^{94} -2.89898 q^{95} -9.79796 q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 2 q^{8} - 2 q^{10} + 2 q^{11} - 4 q^{13} + 4 q^{14} + 2 q^{16} + 4 q^{17} - 4 q^{19} - 2 q^{20} + 2 q^{22} - 2 q^{23} + 4 q^{25} - 4 q^{26} + 4 q^{28} + 8 q^{29} + 6 q^{31} + 2 q^{32} + 4 q^{34} - 4 q^{35} + 10 q^{37} - 4 q^{38} - 2 q^{40} + 6 q^{41} + 20 q^{43} + 2 q^{44} - 2 q^{46} - 6 q^{49} + 4 q^{50} - 4 q^{52} - 10 q^{53} + 10 q^{55} + 4 q^{56} + 8 q^{58} + 12 q^{59} + 26 q^{61} + 6 q^{62} + 2 q^{64} - 20 q^{65} - 2 q^{67} + 4 q^{68} - 4 q^{70} - 18 q^{71} + 12 q^{73} + 10 q^{74} - 4 q^{76} + 4 q^{77} + 12 q^{79} - 2 q^{80} + 6 q^{82} - 10 q^{83} - 4 q^{85} + 20 q^{86} + 2 q^{88} - 8 q^{91} - 2 q^{92} + 4 q^{95} - 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 + 2 * q^8 - 2 * q^10 + 2 * q^11 - 4 * q^13 + 4 * q^14 + 2 * q^16 + 4 * q^17 - 4 * q^19 - 2 * q^20 + 2 * q^22 - 2 * q^23 + 4 * q^25 - 4 * q^26 + 4 * q^28 + 8 * q^29 + 6 * q^31 + 2 * q^32 + 4 * q^34 - 4 * q^35 + 10 * q^37 - 4 * q^38 - 2 * q^40 + 6 * q^41 + 20 * q^43 + 2 * q^44 - 2 * q^46 - 6 * q^49 + 4 * q^50 - 4 * q^52 - 10 * q^53 + 10 * q^55 + 4 * q^56 + 8 * q^58 + 12 * q^59 + 26 * q^61 + 6 * q^62 + 2 * q^64 - 20 * q^65 - 2 * q^67 + 4 * q^68 - 4 * q^70 - 18 * q^71 + 12 * q^73 + 10 * q^74 - 4 * q^76 + 4 * q^77 + 12 * q^79 - 2 * q^80 + 6 * q^82 - 10 * q^83 - 4 * q^85 + 20 * q^86 + 2 * q^88 - 8 * q^91 - 2 * q^92 + 4 * q^95 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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