LMFDB - Embedded newform 3762.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3762,2,Mod(1,3762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3762, 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("3762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3762.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:	$30.0397212404$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	3762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +3.56155 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +3.56155 q^{7} +1.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} -3.56155 q^{13} +3.56155 q^{14} +1.00000 q^{16} -3.56155 q^{17} -1.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} -5.56155 q^{23} -1.00000 q^{25} -3.56155 q^{26} +3.56155 q^{28} -6.68466 q^{29} +2.00000 q^{31} +1.00000 q^{32} -3.56155 q^{34} -7.12311 q^{35} +3.12311 q^{37} -1.00000 q^{38} -2.00000 q^{40} -2.00000 q^{41} -1.00000 q^{44} -5.56155 q^{46} -8.00000 q^{47} +5.68466 q^{49} -1.00000 q^{50} -3.56155 q^{52} +7.80776 q^{53} +2.00000 q^{55} +3.56155 q^{56} -6.68466 q^{58} -4.68466 q^{59} -10.2462 q^{61} +2.00000 q^{62} +1.00000 q^{64} +7.12311 q^{65} +4.68466 q^{67} -3.56155 q^{68} -7.12311 q^{70} -6.00000 q^{71} +2.68466 q^{73} +3.12311 q^{74} -1.00000 q^{76} -3.56155 q^{77} -4.00000 q^{79} -2.00000 q^{80} -2.00000 q^{82} +2.24621 q^{83} +7.12311 q^{85} -1.00000 q^{88} +9.12311 q^{89} -12.6847 q^{91} -5.56155 q^{92} -8.00000 q^{94} +2.00000 q^{95} +1.12311 q^{97} +5.68466 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 3 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8} - 4 q^{10} - 2 q^{11} - 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} - 2 q^{19} - 4 q^{20} - 2 q^{22} - 7 q^{23} - 2 q^{25} - 3 q^{26} + 3 q^{28} - q^{29} + 4 q^{31} + 2 q^{32} - 3 q^{34} - 6 q^{35} - 2 q^{37} - 2 q^{38} - 4 q^{40} - 4 q^{41} - 2 q^{44} - 7 q^{46} - 16 q^{47} - q^{49} - 2 q^{50} - 3 q^{52} - 5 q^{53} + 4 q^{55} + 3 q^{56} - q^{58} + 3 q^{59} - 4 q^{61} + 4 q^{62} + 2 q^{64} + 6 q^{65} - 3 q^{67} - 3 q^{68} - 6 q^{70} - 12 q^{71} - 7 q^{73} - 2 q^{74} - 2 q^{76} - 3 q^{77} - 8 q^{79} - 4 q^{80} - 4 q^{82} - 12 q^{83} + 6 q^{85} - 2 q^{88} + 10 q^{89} - 13 q^{91} - 7 q^{92} - 16 q^{94} + 4 q^{95} - 6 q^{97} - q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 3 * q^7 + 2 * q^8 - 4 * q^10 - 2 * q^11 - 3 * q^13 + 3 * q^14 + 2 * q^16 - 3 * q^17 - 2 * q^19 - 4 * q^20 - 2 * q^22 - 7 * q^23 - 2 * q^25 - 3 * q^26 + 3 * q^28 - q^29 + 4 * q^31 + 2 * q^32 - 3 * q^34 - 6 * q^35 - 2 * q^37 - 2 * q^38 - 4 * q^40 - 4 * q^41 - 2 * q^44 - 7 * q^46 - 16 * q^47 - q^49 - 2 * q^50 - 3 * q^52 - 5 * q^53 + 4 * q^55 + 3 * q^56 - q^58 + 3 * q^59 - 4 * q^61 + 4 * q^62 + 2 * q^64 + 6 * q^65 - 3 * q^67 - 3 * q^68 - 6 * q^70 - 12 * q^71 - 7 * q^73 - 2 * q^74 - 2 * q^76 - 3 * q^77 - 8 * q^79 - 4 * q^80 - 4 * q^82 - 12 * q^83 + 6 * q^85 - 2 * q^88 + 10 * q^89 - 13 * q^91 - 7 * q^92 - 16 * q^94 + 4 * q^95 - 6 * q^97 - 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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	3.56155	1.34614	0.673070	−	0.739579i	$-0.264975\pi$
$7$	3.56155	1.34614	0.673070	+	0.739579i	$0.264975\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.00000	−0.632456
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.56155	−0.987797	−0.493899	−	0.869520i	$-0.664429\pi$
$13$	−3.56155	−0.987797	−0.493899	+	0.869520i	$0.664429\pi$
$14$	3.56155	0.951865
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.56155	−0.863803	−0.431902	−	0.901921i	$-0.642157\pi$
$17$	−3.56155	−0.863803	−0.431902	+	0.901921i	$0.642157\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−5.56155	−1.15966	−0.579832	−	0.814736i	$-0.696882\pi$
$23$	−5.56155	−1.15966	−0.579832	+	0.814736i	$0.696882\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	−3.56155	−0.698478
$27$	0	0
$28$	3.56155	0.673070
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.56155	−0.610801
$35$	−7.12311	−1.20402
$36$	0	0
$37$	3.12311	0.513435	0.256718	−	0.966486i	$-0.417359\pi$
$37$	3.12311	0.513435	0.256718	+	0.966486i	$0.417359\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−2.00000	−0.316228
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−5.56155	−0.820006
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	5.68466	0.812094
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−3.56155	−0.493899
$53$	7.80776	1.07248	0.536239	−	0.844066i	$-0.319844\pi$
$53$	7.80776	1.07248	0.536239	+	0.844066i	$0.319844\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	3.56155	0.475933
$57$	0	0
$58$	−6.68466	−0.877739
$59$	−4.68466	−0.609891	−0.304945	−	0.952370i	$-0.598638\pi$
$59$	−4.68466	−0.609891	−0.304945	+	0.952370i	$0.598638\pi$
$60$	0	0
$61$	−10.2462	−1.31189	−0.655946	−	0.754807i	$-0.727730\pi$
$61$	−10.2462	−1.31189	−0.655946	+	0.754807i	$0.727730\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	7.12311	0.883513
$66$	0	0
$67$	4.68466	0.572322	0.286161	−	0.958182i	$-0.407621\pi$
$67$	4.68466	0.572322	0.286161	+	0.958182i	$0.407621\pi$
$68$	−3.56155	−0.431902
$69$	0	0
$70$	−7.12311	−0.851374
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	2.68466	0.314216	0.157108	−	0.987581i	$-0.449783\pi$
$73$	2.68466	0.314216	0.157108	+	0.987581i	$0.449783\pi$
$74$	3.12311	0.363054
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−3.56155	−0.405877
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	−2.00000	−0.220863
$83$	2.24621	0.246554	0.123277	−	0.992372i	$-0.460660\pi$
$83$	2.24621	0.246554	0.123277	+	0.992372i	$0.460660\pi$
$84$	0	0
$85$	7.12311	0.772609
$86$	0	0
$87$	0	0
$88$	−1.00000	−0.106600
$89$	9.12311	0.967047	0.483524	−	0.875331i	$-0.339357\pi$
$89$	9.12311	0.967047	0.483524	+	0.875331i	$0.339357\pi$
$90$	0	0
$91$	−12.6847	−1.32971
$92$	−5.56155	−0.579832
$93$	0	0
$94$	−8.00000	−0.825137
$95$	2.00000	0.205196
$96$	0	0
$97$	1.12311	0.114034	0.0570170	−	0.998373i	$-0.481841\pi$
$97$	1.12311	0.114034	0.0570170	+	0.998373i	$0.481841\pi$
$98$	5.68466	0.574237
$99$	0	0
$100$	−1.00000	−0.100000
$101$	15.1231	1.50481	0.752403	−	0.658703i	$-0.228895\pi$
$101$	15.1231	1.50481	0.752403	+	0.658703i	$0.228895\pi$
$102$	0	0
$103$	11.3693	1.12025	0.560126	−	0.828407i	$-0.310753\pi$
$103$	11.3693	1.12025	0.560126	+	0.828407i	$0.310753\pi$
$104$	−3.56155	−0.349239
$105$	0	0
$106$	7.80776	0.758357
$107$	−18.9309	−1.83012	−0.915058	−	0.403322i	$-0.867855\pi$
$107$	−18.9309	−1.83012	−0.915058	+	0.403322i	$0.867855\pi$
$108$	0	0
$109$	−18.6847	−1.78967	−0.894833	−	0.446401i	$-0.852705\pi$
$109$	−18.6847	−1.78967	−0.894833	+	0.446401i	$0.852705\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	3.56155	0.336535
$113$	−5.12311	−0.481941	−0.240971	−	0.970532i	$-0.577466\pi$
$113$	−5.12311	−0.481941	−0.240971	+	0.970532i	$0.577466\pi$
$114$	0	0
$115$	11.1231	1.03723
$116$	−6.68466	−0.620655
$117$	0	0
$118$	−4.68466	−0.431258
$119$	−12.6847	−1.16280
$120$	0	0
$121$	1.00000	0.0909091
$122$	−10.2462	−0.927648
$123$	0	0
$124$	2.00000	0.179605
$125$	12.0000	1.07331
$126$	0	0
$127$	−14.2462	−1.26415	−0.632073	−	0.774909i	$-0.717796\pi$
$127$	−14.2462	−1.26415	−0.632073	+	0.774909i	$0.717796\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	7.12311	0.624738
$131$	7.12311	0.622349	0.311174	−	0.950353i	$-0.399278\pi$
$131$	7.12311	0.622349	0.311174	+	0.950353i	$0.399278\pi$
$132$	0	0
$133$	−3.56155	−0.308826
$134$	4.68466	0.404693
$135$	0	0
$136$	−3.56155	−0.305401
$137$	−8.43845	−0.720945	−0.360473	−	0.932770i	$-0.617385\pi$
$137$	−8.43845	−0.720945	−0.360473	+	0.932770i	$0.617385\pi$
$138$	0	0
$139$	−17.3693	−1.47325	−0.736623	−	0.676303i	$-0.763581\pi$
$139$	−17.3693	−1.47325	−0.736623	+	0.676303i	$0.763581\pi$
$140$	−7.12311	−0.602012
$141$	0	0
$142$	−6.00000	−0.503509
$143$	3.56155	0.297832
$144$	0	0
$145$	13.3693	1.11026
$146$	2.68466	0.222184
$147$	0	0
$148$	3.12311	0.256718
$149$	15.1231	1.23893	0.619467	−	0.785023i	$-0.287349\pi$
$149$	15.1231	1.23893	0.619467	+	0.785023i	$0.287349\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−3.56155	−0.286998
$155$	−4.00000	−0.321288
$156$	0	0
$157$	18.4924	1.47586	0.737928	−	0.674879i	$-0.235804\pi$
$157$	18.4924	1.47586	0.737928	+	0.674879i	$0.235804\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	−2.00000	−0.158114
$161$	−19.8078	−1.56107
$162$	0	0
$163$	−13.3693	−1.04717	−0.523583	−	0.851975i	$-0.675405\pi$
$163$	−13.3693	−1.04717	−0.523583	+	0.851975i	$0.675405\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	2.24621	0.174340
$167$	25.3693	1.96314	0.981568	−	0.191111i	$-0.0612092\pi$
$167$	25.3693	1.96314	0.981568	+	0.191111i	$0.0612092\pi$
$168$	0	0
$169$	−0.315342	−0.0242570
$170$	7.12311	0.546317
$171$	0	0
$172$	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
$175$	−3.56155	−0.269228
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	9.12311	0.683806
$179$	−22.2462	−1.66276	−0.831380	−	0.555704i	$-0.812449\pi$
$179$	−22.2462	−1.66276	−0.831380	+	0.555704i	$0.812449\pi$
$180$	0	0
$181$	−19.1231	−1.42141	−0.710705	−	0.703491i	$-0.751624\pi$
$181$	−19.1231	−1.42141	−0.710705	+	0.703491i	$0.751624\pi$
$182$	−12.6847	−0.940249
$183$	0	0
$184$	−5.56155	−0.410003
$185$	−6.24621	−0.459231
$186$	0	0
$187$	3.56155	0.260447
$188$	−8.00000	−0.583460
$189$	0	0
$190$	2.00000	0.145095
$191$	20.6847	1.49669	0.748345	−	0.663310i	$-0.230849\pi$
$191$	20.6847	1.49669	0.748345	+	0.663310i	$0.230849\pi$
$192$	0	0
$193$	−1.12311	−0.0808429	−0.0404215	−	0.999183i	$-0.512870\pi$
$193$	−1.12311	−0.0808429	−0.0404215	+	0.999183i	$0.512870\pi$
$194$	1.12311	0.0806343
$195$	0	0
$196$	5.68466	0.406047
$197$	−1.75379	−0.124952	−0.0624761	−	0.998046i	$-0.519900\pi$
$197$	−1.75379	−0.124952	−0.0624761	+	0.998046i	$0.519900\pi$
$198$	0	0
$199$	−0.684658	−0.0485341	−0.0242671	−	0.999706i	$-0.507725\pi$
$199$	−0.684658	−0.0485341	−0.0242671	+	0.999706i	$0.507725\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	15.1231	1.06406
$203$	−23.8078	−1.67098
$204$	0	0
$205$	4.00000	0.279372
$206$	11.3693	0.792138
$207$	0	0
$208$	−3.56155	−0.246949
$209$	1.00000	0.0691714
$210$	0	0
$211$	−12.6847	−0.873248	−0.436624	−	0.899644i	$-0.643826\pi$
$211$	−12.6847	−0.873248	−0.436624	+	0.899644i	$0.643826\pi$
$212$	7.80776	0.536239
$213$	0	0
$214$	−18.9309	−1.29409
$215$	0	0
$216$	0	0
$217$	7.12311	0.483548
$218$	−18.6847	−1.26548
$219$	0	0
$220$	2.00000	0.134840
$221$	12.6847	0.853262
$222$	0	0
$223$	−24.7386	−1.65662	−0.828311	−	0.560269i	$-0.810698\pi$
$223$	−24.7386	−1.65662	−0.828311	+	0.560269i	$0.810698\pi$
$224$	3.56155	0.237966
$225$	0	0
$226$	−5.12311	−0.340784
$227$	17.1771	1.14008	0.570041	−	0.821616i	$-0.306927\pi$
$227$	17.1771	1.14008	0.570041	+	0.821616i	$0.306927\pi$
$228$	0	0
$229$	21.1231	1.39585	0.697927	−	0.716169i	$-0.254106\pi$
$229$	21.1231	1.39585	0.697927	+	0.716169i	$0.254106\pi$
$230$	11.1231	0.733436
$231$	0	0
$232$	−6.68466	−0.438869
$233$	−20.7386	−1.35863	−0.679317	−	0.733845i	$-0.737724\pi$
$233$	−20.7386	−1.35863	−0.679317	+	0.733845i	$0.737724\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	−4.68466	−0.304945
$237$	0	0
$238$	−12.6847	−0.822224
$239$	4.93087	0.318951	0.159476	−	0.987202i	$-0.449020\pi$
$239$	4.93087	0.318951	0.159476	+	0.987202i	$0.449020\pi$
$240$	0	0
$241$	21.6155	1.39238	0.696189	−	0.717858i	$-0.254877\pi$
$241$	21.6155	1.39238	0.696189	+	0.717858i	$0.254877\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−10.2462	−0.655946
$245$	−11.3693	−0.726359
$246$	0	0
$247$	3.56155	0.226616
$248$	2.00000	0.127000
$249$	0	0
$250$	12.0000	0.758947
$251$	−8.87689	−0.560305	−0.280152	−	0.959956i	$-0.590385\pi$
$251$	−8.87689	−0.560305	−0.280152	+	0.959956i	$0.590385\pi$
$252$	0	0
$253$	5.56155	0.349652
$254$	−14.2462	−0.893887
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	11.1231	0.691156
$260$	7.12311	0.441756
$261$	0	0
$262$	7.12311	0.440067
$263$	−17.1231	−1.05586	−0.527928	−	0.849289i	$-0.677031\pi$
$263$	−17.1231	−1.05586	−0.527928	+	0.849289i	$0.677031\pi$
$264$	0	0
$265$	−15.6155	−0.959254
$266$	−3.56155	−0.218373
$267$	0	0
$268$	4.68466	0.286161
$269$	−11.1231	−0.678188	−0.339094	−	0.940753i	$-0.610120\pi$
$269$	−11.1231	−0.678188	−0.339094	+	0.940753i	$0.610120\pi$
$270$	0	0
$271$	13.3153	0.808849	0.404425	−	0.914571i	$-0.367472\pi$
$271$	13.3153	0.808849	0.404425	+	0.914571i	$0.367472\pi$
$272$	−3.56155	−0.215951
$273$	0	0
$274$	−8.43845	−0.509785
$275$	1.00000	0.0603023
$276$	0	0
$277$	−24.4924	−1.47161	−0.735804	−	0.677195i	$-0.763195\pi$
$277$	−24.4924	−1.47161	−0.735804	+	0.677195i	$0.763195\pi$
$278$	−17.3693	−1.04174
$279$	0	0
$280$	−7.12311	−0.425687
$281$	0.630683	0.0376234	0.0188117	−	0.999823i	$-0.494012\pi$
$281$	0.630683	0.0376234	0.0188117	+	0.999823i	$0.494012\pi$
$282$	0	0
$283$	−13.3693	−0.794723	−0.397362	−	0.917662i	$-0.630074\pi$
$283$	−13.3693	−0.794723	−0.397362	+	0.917662i	$0.630074\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	3.56155	0.210599
$287$	−7.12311	−0.420464
$288$	0	0
$289$	−4.31534	−0.253844
$290$	13.3693	0.785073
$291$	0	0
$292$	2.68466	0.157108
$293$	24.9309	1.45648	0.728238	−	0.685324i	$-0.240339\pi$
$293$	24.9309	1.45648	0.728238	+	0.685324i	$0.240339\pi$
$294$	0	0
$295$	9.36932	0.545503
$296$	3.12311	0.181527
$297$	0	0
$298$	15.1231	0.876058
$299$	19.8078	1.14551
$300$	0	0
$301$	0	0
$302$	4.00000	0.230174
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	20.4924	1.17339
$306$	0	0
$307$	−5.75379	−0.328386	−0.164193	−	0.986428i	$-0.552502\pi$
$307$	−5.75379	−0.328386	−0.164193	+	0.986428i	$0.552502\pi$
$308$	−3.56155	−0.202938
$309$	0	0
$310$	−4.00000	−0.227185
$311$	−15.8078	−0.896376	−0.448188	−	0.893939i	$-0.647930\pi$
$311$	−15.8078	−0.896376	−0.448188	+	0.893939i	$0.647930\pi$
$312$	0	0
$313$	3.56155	0.201311	0.100655	−	0.994921i	$-0.467906\pi$
$313$	3.56155	0.201311	0.100655	+	0.994921i	$0.467906\pi$
$314$	18.4924	1.04359
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−30.0540	−1.68800	−0.844000	−	0.536344i	$-0.819805\pi$
$317$	−30.0540	−1.68800	−0.844000	+	0.536344i	$0.819805\pi$
$318$	0	0
$319$	6.68466	0.374269
$320$	−2.00000	−0.111803
$321$	0	0
$322$	−19.8078	−1.10384
$323$	3.56155	0.198170
$324$	0	0
$325$	3.56155	0.197559
$326$	−13.3693	−0.740458
$327$	0	0
$328$	−2.00000	−0.110432
$329$	−28.4924	−1.57084
$330$	0	0
$331$	−10.4384	−0.573749	−0.286874	−	0.957968i	$-0.592616\pi$
$331$	−10.4384	−0.573749	−0.286874	+	0.957968i	$0.592616\pi$
$332$	2.24621	0.123277
$333$	0	0
$334$	25.3693	1.38815
$335$	−9.36932	−0.511900
$336$	0	0
$337$	2.87689	0.156714	0.0783572	−	0.996925i	$-0.475033\pi$
$337$	2.87689	0.156714	0.0783572	+	0.996925i	$0.475033\pi$
$338$	−0.315342	−0.0171523
$339$	0	0
$340$	7.12311	0.386305
$341$	−2.00000	−0.108306
$342$	0	0
$343$	−4.68466	−0.252948
$344$	0	0
$345$	0	0
$346$	−18.0000	−0.967686
$347$	31.6155	1.69721	0.848605	−	0.529027i	$-0.177443\pi$
$347$	31.6155	1.69721	0.848605	+	0.529027i	$0.177443\pi$
$348$	0	0
$349$	−19.6155	−1.05000	−0.524998	−	0.851104i	$-0.675934\pi$
$349$	−19.6155	−1.05000	−0.524998	+	0.851104i	$0.675934\pi$
$350$	−3.56155	−0.190373
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	23.1771	1.23359	0.616796	−	0.787123i	$-0.288430\pi$
$353$	23.1771	1.23359	0.616796	+	0.787123i	$0.288430\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	9.12311	0.483524
$357$	0	0
$358$	−22.2462	−1.17575
$359$	−12.9309	−0.682465	−0.341233	−	0.939979i	$-0.610844\pi$
$359$	−12.9309	−0.682465	−0.341233	+	0.939979i	$0.610844\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−19.1231	−1.00509
$363$	0	0
$364$	−12.6847	−0.664857
$365$	−5.36932	−0.281043
$366$	0	0
$367$	13.7538	0.717942	0.358971	−	0.933349i	$-0.383128\pi$
$367$	13.7538	0.717942	0.358971	+	0.933349i	$0.383128\pi$
$368$	−5.56155	−0.289916
$369$	0	0
$370$	−6.24621	−0.324725
$371$	27.8078	1.44371
$372$	0	0
$373$	7.56155	0.391522	0.195761	−	0.980652i	$-0.437282\pi$
$373$	7.56155	0.391522	0.195761	+	0.980652i	$0.437282\pi$
$374$	3.56155	0.184164
$375$	0	0
$376$	−8.00000	−0.412568
$377$	23.8078	1.22616
$378$	0	0
$379$	2.43845	0.125255	0.0626273	−	0.998037i	$-0.480052\pi$
$379$	2.43845	0.125255	0.0626273	+	0.998037i	$0.480052\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	20.6847	1.05832
$383$	2.00000	0.102195	0.0510976	−	0.998694i	$-0.483728\pi$
$383$	2.00000	0.102195	0.0510976	+	0.998694i	$0.483728\pi$
$384$	0	0
$385$	7.12311	0.363027
$386$	−1.12311	−0.0571646
$387$	0	0
$388$	1.12311	0.0570170
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	19.8078	1.00172
$392$	5.68466	0.287119
$393$	0	0
$394$	−1.75379	−0.0883546
$395$	8.00000	0.402524
$396$	0	0
$397$	−26.9848	−1.35433	−0.677165	−	0.735831i	$-0.736792\pi$
$397$	−26.9848	−1.35433	−0.677165	+	0.735831i	$0.736792\pi$
$398$	−0.684658	−0.0343188
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	24.2462	1.21080	0.605399	−	0.795922i	$-0.293014\pi$
$401$	24.2462	1.21080	0.605399	+	0.795922i	$0.293014\pi$
$402$	0	0
$403$	−7.12311	−0.354827
$404$	15.1231	0.752403
$405$	0	0
$406$	−23.8078	−1.18156
$407$	−3.12311	−0.154807
$408$	0	0
$409$	−0.246211	−0.0121744	−0.00608718	−	0.999981i	$-0.501938\pi$
$409$	−0.246211	−0.0121744	−0.00608718	+	0.999981i	$0.501938\pi$
$410$	4.00000	0.197546
$411$	0	0
$412$	11.3693	0.560126
$413$	−16.6847	−0.820998
$414$	0	0
$415$	−4.49242	−0.220524
$416$	−3.56155	−0.174619
$417$	0	0
$418$	1.00000	0.0489116
$419$	23.1231	1.12964	0.564819	−	0.825215i	$-0.308946\pi$
$419$	23.1231	1.12964	0.564819	+	0.825215i	$0.308946\pi$
$420$	0	0
$421$	−1.06913	−0.0521062	−0.0260531	−	0.999661i	$-0.508294\pi$
$421$	−1.06913	−0.0521062	−0.0260531	+	0.999661i	$0.508294\pi$
$422$	−12.6847	−0.617480
$423$	0	0
$424$	7.80776	0.379179
$425$	3.56155	0.172761
$426$	0	0
$427$	−36.4924	−1.76599
$428$	−18.9309	−0.915058
$429$	0	0
$430$	0	0
$431$	25.8617	1.24572	0.622858	−	0.782335i	$-0.285971\pi$
$431$	25.8617	1.24572	0.622858	+	0.782335i	$0.285971\pi$
$432$	0	0
$433$	31.3693	1.50751	0.753757	−	0.657154i	$-0.228240\pi$
$433$	31.3693	1.50751	0.753757	+	0.657154i	$0.228240\pi$
$434$	7.12311	0.341920
$435$	0	0
$436$	−18.6847	−0.894833
$437$	5.56155	0.266045
$438$	0	0
$439$	31.6155	1.50893	0.754463	−	0.656342i	$-0.227897\pi$
$439$	31.6155	1.50893	0.754463	+	0.656342i	$0.227897\pi$
$440$	2.00000	0.0953463
$441$	0	0
$442$	12.6847	0.603348
$443$	17.8617	0.848637	0.424318	−	0.905513i	$-0.360514\pi$
$443$	17.8617	0.848637	0.424318	+	0.905513i	$0.360514\pi$
$444$	0	0
$445$	−18.2462	−0.864953
$446$	−24.7386	−1.17141
$447$	0	0
$448$	3.56155	0.168268
$449$	17.6155	0.831328	0.415664	−	0.909518i	$-0.363549\pi$
$449$	17.6155	0.831328	0.415664	+	0.909518i	$0.363549\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	−5.12311	−0.240971
$453$	0	0
$454$	17.1771	0.806160
$455$	25.3693	1.18933
$456$	0	0
$457$	32.4384	1.51741	0.758703	−	0.651436i	$-0.225833\pi$
$457$	32.4384	1.51741	0.758703	+	0.651436i	$0.225833\pi$
$458$	21.1231	0.987018
$459$	0	0
$460$	11.1231	0.518617
$461$	22.7386	1.05904	0.529522	−	0.848296i	$-0.322371\pi$
$461$	22.7386	1.05904	0.529522	+	0.848296i	$0.322371\pi$
$462$	0	0
$463$	36.4924	1.69595	0.847973	−	0.530039i	$-0.177823\pi$
$463$	36.4924	1.69595	0.847973	+	0.530039i	$0.177823\pi$
$464$	−6.68466	−0.310327
$465$	0	0
$466$	−20.7386	−0.960699
$467$	−26.2462	−1.21453	−0.607265	−	0.794499i	$-0.707733\pi$
$467$	−26.2462	−1.21453	−0.607265	+	0.794499i	$0.707733\pi$
$468$	0	0
$469$	16.6847	0.770426
$470$	16.0000	0.738025
$471$	0	0
$472$	−4.68466	−0.215629
$473$	0	0
$474$	0	0
$475$	1.00000	0.0458831
$476$	−12.6847	−0.581400
$477$	0	0
$478$	4.93087	0.225533
$479$	−11.3693	−0.519477	−0.259739	−	0.965679i	$-0.583636\pi$
$479$	−11.3693	−0.519477	−0.259739	+	0.965679i	$0.583636\pi$
$480$	0	0
$481$	−11.1231	−0.507170
$482$	21.6155	0.984560
$483$	0	0
$484$	1.00000	0.0454545
$485$	−2.24621	−0.101995
$486$	0	0
$487$	22.8769	1.03665	0.518326	−	0.855183i	$-0.326556\pi$
$487$	22.8769	1.03665	0.518326	+	0.855183i	$0.326556\pi$
$488$	−10.2462	−0.463824
$489$	0	0
$490$	−11.3693	−0.513613
$491$	14.6307	0.660273	0.330137	−	0.943933i	$-0.392905\pi$
$491$	14.6307	0.660273	0.330137	+	0.943933i	$0.392905\pi$
$492$	0	0
$493$	23.8078	1.07225
$494$	3.56155	0.160242
$495$	0	0
$496$	2.00000	0.0898027
$497$	−21.3693	−0.958545
$498$	0	0
$499$	26.2462	1.17494	0.587471	−	0.809245i	$-0.300124\pi$
$499$	26.2462	1.17494	0.587471	+	0.809245i	$0.300124\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	−8.87689	−0.396195
$503$	−9.31534	−0.415351	−0.207675	−	0.978198i	$-0.566590\pi$
$503$	−9.31534	−0.415351	−0.207675	+	0.978198i	$0.566590\pi$
$504$	0	0
$505$	−30.2462	−1.34594
$506$	5.56155	0.247241
$507$	0	0
$508$	−14.2462	−0.632073
$509$	6.63068	0.293900	0.146950	−	0.989144i	$-0.453054\pi$
$509$	6.63068	0.293900	0.146950	+	0.989144i	$0.453054\pi$
$510$	0	0
$511$	9.56155	0.422978
$512$	1.00000	0.0441942
$513$	0	0
$514$	22.0000	0.970378
$515$	−22.7386	−1.00198
$516$	0	0
$517$	8.00000	0.351840
$518$	11.1231	0.488721
$519$	0	0
$520$	7.12311	0.312369
$521$	−1.12311	−0.0492042	−0.0246021	−	0.999697i	$-0.507832\pi$
$521$	−1.12311	−0.0492042	−0.0246021	+	0.999697i	$0.507832\pi$
$522$	0	0
$523$	−6.05398	−0.264722	−0.132361	−	0.991202i	$-0.542256\pi$
$523$	−6.05398	−0.264722	−0.132361	+	0.991202i	$0.542256\pi$
$524$	7.12311	0.311174
$525$	0	0
$526$	−17.1231	−0.746603
$527$	−7.12311	−0.310287
$528$	0	0
$529$	7.93087	0.344820
$530$	−15.6155	−0.678295
$531$	0	0
$532$	−3.56155	−0.154413
$533$	7.12311	0.308536
$534$	0	0
$535$	37.8617	1.63691
$536$	4.68466	0.202346
$537$	0	0
$538$	−11.1231	−0.479551
$539$	−5.68466	−0.244856
$540$	0	0
$541$	43.2311	1.85865	0.929324	−	0.369265i	$-0.120391\pi$
$541$	43.2311	1.85865	0.929324	+	0.369265i	$0.120391\pi$
$542$	13.3153	0.571943
$543$	0	0
$544$	−3.56155	−0.152700
$545$	37.3693	1.60073
$546$	0	0
$547$	6.73863	0.288123	0.144062	−	0.989569i	$-0.453984\pi$
$547$	6.73863	0.288123	0.144062	+	0.989569i	$0.453984\pi$
$548$	−8.43845	−0.360473
$549$	0	0
$550$	1.00000	0.0426401
$551$	6.68466	0.284776
$552$	0	0
$553$	−14.2462	−0.605811
$554$	−24.4924	−1.04058
$555$	0	0
$556$	−17.3693	−0.736623
$557$	−8.87689	−0.376126	−0.188063	−	0.982157i	$-0.560221\pi$
$557$	−8.87689	−0.376126	−0.188063	+	0.982157i	$0.560221\pi$
$558$	0	0
$559$	0	0
$560$	−7.12311	−0.301006
$561$	0	0
$562$	0.630683	0.0266038
$563$	−6.73863	−0.284000	−0.142000	−	0.989867i	$-0.545353\pi$
$563$	−6.73863	−0.284000	−0.142000	+	0.989867i	$0.545353\pi$
$564$	0	0
$565$	10.2462	0.431061
$566$	−13.3693	−0.561954
$567$	0	0
$568$	−6.00000	−0.251754
$569$	−3.36932	−0.141249	−0.0706246	−	0.997503i	$-0.522499\pi$
$569$	−3.36932	−0.141249	−0.0706246	+	0.997503i	$0.522499\pi$
$570$	0	0
$571$	35.6155	1.49046	0.745232	−	0.666806i	$-0.232339\pi$
$571$	35.6155	1.49046	0.745232	+	0.666806i	$0.232339\pi$
$572$	3.56155	0.148916
$573$	0	0
$574$	−7.12311	−0.297313
$575$	5.56155	0.231933
$576$	0	0
$577$	3.94602	0.164275	0.0821376	−	0.996621i	$-0.473825\pi$
$577$	3.94602	0.164275	0.0821376	+	0.996621i	$0.473825\pi$
$578$	−4.31534	−0.179495
$579$	0	0
$580$	13.3693	0.555131
$581$	8.00000	0.331896
$582$	0	0
$583$	−7.80776	−0.323365
$584$	2.68466	0.111092
$585$	0	0
$586$	24.9309	1.02988
$587$	7.50758	0.309871	0.154935	−	0.987925i	$-0.450483\pi$
$587$	7.50758	0.309871	0.154935	+	0.987925i	$0.450483\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	9.36932	0.385729
$591$	0	0
$592$	3.12311	0.128359
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	25.3693	1.04004
$596$	15.1231	0.619467
$597$	0	0
$598$	19.8078	0.810000
$599$	−39.8617	−1.62871	−0.814353	−	0.580370i	$-0.802908\pi$
$599$	−39.8617	−1.62871	−0.814353	+	0.580370i	$0.802908\pi$
$600$	0	0
$601$	−3.36932	−0.137437	−0.0687187	−	0.997636i	$-0.521891\pi$
$601$	−3.36932	−0.137437	−0.0687187	+	0.997636i	$0.521891\pi$
$602$	0	0
$603$	0	0
$604$	4.00000	0.162758
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	27.6155	1.12088	0.560440	−	0.828195i	$-0.310632\pi$
$607$	27.6155	1.12088	0.560440	+	0.828195i	$0.310632\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	20.4924	0.829714
$611$	28.4924	1.15268
$612$	0	0
$613$	1.75379	0.0708349	0.0354174	−	0.999373i	$-0.488724\pi$
$613$	1.75379	0.0708349	0.0354174	+	0.999373i	$0.488724\pi$
$614$	−5.75379	−0.232204
$615$	0	0
$616$	−3.56155	−0.143499
$617$	4.73863	0.190770	0.0953851	−	0.995440i	$-0.469592\pi$
$617$	4.73863	0.190770	0.0953851	+	0.995440i	$0.469592\pi$
$618$	0	0
$619$	−26.2462	−1.05492	−0.527462	−	0.849579i	$-0.676856\pi$
$619$	−26.2462	−1.05492	−0.527462	+	0.849579i	$0.676856\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	−15.8078	−0.633834
$623$	32.4924	1.30178
$624$	0	0
$625$	−19.0000	−0.760000
$626$	3.56155	0.142348
$627$	0	0
$628$	18.4924	0.737928
$629$	−11.1231	−0.443507
$630$	0	0
$631$	36.4924	1.45274	0.726370	−	0.687304i	$-0.241206\pi$
$631$	36.4924	1.45274	0.726370	+	0.687304i	$0.241206\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	−30.0540	−1.19360
$635$	28.4924	1.13069
$636$	0	0
$637$	−20.2462	−0.802184
$638$	6.68466	0.264648
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	9.61553	0.379791	0.189895	−	0.981804i	$-0.439185\pi$
$641$	9.61553	0.379791	0.189895	+	0.981804i	$0.439185\pi$
$642$	0	0
$643$	−37.3693	−1.47370	−0.736851	−	0.676055i	$-0.763688\pi$
$643$	−37.3693	−1.47370	−0.736851	+	0.676055i	$0.763688\pi$
$644$	−19.8078	−0.780535
$645$	0	0
$646$	3.56155	0.140127
$647$	−22.5464	−0.886390	−0.443195	−	0.896425i	$-0.646155\pi$
$647$	−22.5464	−0.886390	−0.443195	+	0.896425i	$0.646155\pi$
$648$	0	0
$649$	4.68466	0.183889
$650$	3.56155	0.139696
$651$	0	0
$652$	−13.3693	−0.523583
$653$	−25.6155	−1.00241	−0.501207	−	0.865328i	$-0.667110\pi$
$653$	−25.6155	−1.00241	−0.501207	+	0.865328i	$0.667110\pi$
$654$	0	0
$655$	−14.2462	−0.556646
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	−28.4924	−1.11075
$659$	−47.4233	−1.84735	−0.923675	−	0.383178i	$-0.874830\pi$
$659$	−47.4233	−1.84735	−0.923675	+	0.383178i	$0.874830\pi$
$660$	0	0
$661$	−22.4384	−0.872754	−0.436377	−	0.899764i	$-0.643739\pi$
$661$	−22.4384	−0.872754	−0.436377	+	0.899764i	$0.643739\pi$
$662$	−10.4384	−0.405702
$663$	0	0
$664$	2.24621	0.0871699
$665$	7.12311	0.276222
$666$	0	0
$667$	37.1771	1.43950
$668$	25.3693	0.981568
$669$	0	0
$670$	−9.36932	−0.361968
$671$	10.2462	0.395551
$672$	0	0
$673$	−18.8769	−0.727651	−0.363825	−	0.931467i	$-0.618530\pi$
$673$	−18.8769	−0.727651	−0.363825	+	0.931467i	$0.618530\pi$
$674$	2.87689	0.110814
$675$	0	0
$676$	−0.315342	−0.0121285
$677$	39.1771	1.50570	0.752849	−	0.658194i	$-0.228679\pi$
$677$	39.1771	1.50570	0.752849	+	0.658194i	$0.228679\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	7.12311	0.273159
$681$	0	0
$682$	−2.00000	−0.0765840
$683$	4.49242	0.171898	0.0859489	−	0.996300i	$-0.472608\pi$
$683$	4.49242	0.171898	0.0859489	+	0.996300i	$0.472608\pi$
$684$	0	0
$685$	16.8769	0.644833
$686$	−4.68466	−0.178861
$687$	0	0
$688$	0	0
$689$	−27.8078	−1.05939
$690$	0	0
$691$	−43.6155	−1.65921	−0.829606	−	0.558349i	$-0.811435\pi$
$691$	−43.6155	−1.65921	−0.829606	+	0.558349i	$0.811435\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	31.6155	1.20011
$695$	34.7386	1.31771
$696$	0	0
$697$	7.12311	0.269807
$698$	−19.6155	−0.742459
$699$	0	0
$700$	−3.56155	−0.134614
$701$	−24.9848	−0.943665	−0.471832	−	0.881688i	$-0.656407\pi$
$701$	−24.9848	−0.943665	−0.471832	+	0.881688i	$0.656407\pi$
$702$	0	0
$703$	−3.12311	−0.117790
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	23.1771	0.872281
$707$	53.8617	2.02568
$708$	0	0
$709$	−9.50758	−0.357065	−0.178532	−	0.983934i	$-0.557135\pi$
$709$	−9.50758	−0.357065	−0.178532	+	0.983934i	$0.557135\pi$
$710$	12.0000	0.450352
$711$	0	0
$712$	9.12311	0.341903
$713$	−11.1231	−0.416564
$714$	0	0
$715$	−7.12311	−0.266389
$716$	−22.2462	−0.831380
$717$	0	0
$718$	−12.9309	−0.482576
$719$	6.43845	0.240114	0.120057	−	0.992767i	$-0.461692\pi$
$719$	6.43845	0.240114	0.120057	+	0.992767i	$0.461692\pi$
$720$	0	0
$721$	40.4924	1.50802
$722$	1.00000	0.0372161
$723$	0	0
$724$	−19.1231	−0.710705
$725$	6.68466	0.248262
$726$	0	0
$727$	−39.4233	−1.46213	−0.731064	−	0.682308i	$-0.760976\pi$
$727$	−39.4233	−1.46213	−0.731064	+	0.682308i	$0.760976\pi$
$728$	−12.6847	−0.470125
$729$	0	0
$730$	−5.36932	−0.198727
$731$	0	0
$732$	0	0
$733$	−48.1080	−1.77691	−0.888454	−	0.458966i	$-0.848220\pi$
$733$	−48.1080	−1.77691	−0.888454	+	0.458966i	$0.848220\pi$
$734$	13.7538	0.507662
$735$	0	0
$736$	−5.56155	−0.205002
$737$	−4.68466	−0.172562
$738$	0	0
$739$	20.9848	0.771940	0.385970	−	0.922511i	$-0.373867\pi$
$739$	20.9848	0.771940	0.385970	+	0.922511i	$0.373867\pi$
$740$	−6.24621	−0.229615
$741$	0	0
$742$	27.8078	1.02086
$743$	25.7538	0.944815	0.472407	−	0.881380i	$-0.343385\pi$
$743$	25.7538	0.944815	0.472407	+	0.881380i	$0.343385\pi$
$744$	0	0
$745$	−30.2462	−1.10814
$746$	7.56155	0.276848
$747$	0	0
$748$	3.56155	0.130223
$749$	−67.4233	−2.46359
$750$	0	0
$751$	−41.2311	−1.50454	−0.752271	−	0.658853i	$-0.771042\pi$
$751$	−41.2311	−1.50454	−0.752271	+	0.658853i	$0.771042\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	23.8078	0.867028
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−9.50758	−0.345559	−0.172779	−	0.984961i	$-0.555275\pi$
$757$	−9.50758	−0.345559	−0.172779	+	0.984961i	$0.555275\pi$
$758$	2.43845	0.0885684
$759$	0	0
$760$	2.00000	0.0725476
$761$	22.1922	0.804468	0.402234	−	0.915537i	$-0.368234\pi$
$761$	22.1922	0.804468	0.402234	+	0.915537i	$0.368234\pi$
$762$	0	0
$763$	−66.5464	−2.40914
$764$	20.6847	0.748345
$765$	0	0
$766$	2.00000	0.0722629
$767$	16.6847	0.602448
$768$	0	0
$769$	−1.31534	−0.0474324	−0.0237162	−	0.999719i	$-0.507550\pi$
$769$	−1.31534	−0.0474324	−0.0237162	+	0.999719i	$0.507550\pi$
$770$	7.12311	0.256699
$771$	0	0
$772$	−1.12311	−0.0404215
$773$	−9.06913	−0.326194	−0.163097	−	0.986610i	$-0.552148\pi$
$773$	−9.06913	−0.326194	−0.163097	+	0.986610i	$0.552148\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	1.12311	0.0403171
$777$	0	0
$778$	−18.0000	−0.645331
$779$	2.00000	0.0716574
$780$	0	0
$781$	6.00000	0.214697
$782$	19.8078	0.708324
$783$	0	0
$784$	5.68466	0.203024
$785$	−36.9848	−1.32005
$786$	0	0
$787$	3.31534	0.118179	0.0590896	−	0.998253i	$-0.481180\pi$
$787$	3.31534	0.118179	0.0590896	+	0.998253i	$0.481180\pi$
$788$	−1.75379	−0.0624761
$789$	0	0
$790$	8.00000	0.284627
$791$	−18.2462	−0.648761
$792$	0	0
$793$	36.4924	1.29588
$794$	−26.9848	−0.957656
$795$	0	0
$796$	−0.684658	−0.0242671
$797$	35.8078	1.26838	0.634188	−	0.773179i	$-0.281335\pi$
$797$	35.8078	1.26838	0.634188	+	0.773179i	$0.281335\pi$
$798$	0	0
$799$	28.4924	1.00799
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	24.2462	0.856163
$803$	−2.68466	−0.0947395
$804$	0	0
$805$	39.6155	1.39626
$806$	−7.12311	−0.250901
$807$	0	0
$808$	15.1231	0.532029
$809$	−36.9309	−1.29842	−0.649210	−	0.760609i	$-0.724900\pi$
$809$	−36.9309	−1.29842	−0.649210	+	0.760609i	$0.724900\pi$
$810$	0	0
$811$	20.6847	0.726337	0.363168	−	0.931724i	$-0.381695\pi$
$811$	20.6847	0.726337	0.363168	+	0.931724i	$0.381695\pi$
$812$	−23.8078	−0.835489
$813$	0	0
$814$	−3.12311	−0.109465
$815$	26.7386	0.936613
$816$	0	0
$817$	0	0
$818$	−0.246211	−0.00860857
$819$	0	0
$820$	4.00000	0.139686
$821$	−44.9848	−1.56998	−0.784991	−	0.619507i	$-0.787332\pi$
$821$	−44.9848	−1.56998	−0.784991	+	0.619507i	$0.787332\pi$
$822$	0	0
$823$	14.9309	0.520457	0.260229	−	0.965547i	$-0.416202\pi$
$823$	14.9309	0.520457	0.260229	+	0.965547i	$0.416202\pi$
$824$	11.3693	0.396069
$825$	0	0
$826$	−16.6847	−0.580534
$827$	−21.0691	−0.732645	−0.366323	−	0.930488i	$-0.619383\pi$
$827$	−21.0691	−0.732645	−0.366323	+	0.930488i	$0.619383\pi$
$828$	0	0
$829$	−33.1771	−1.15229	−0.576144	−	0.817348i	$-0.695443\pi$
$829$	−33.1771	−1.15229	−0.576144	+	0.817348i	$0.695443\pi$
$830$	−4.49242	−0.155934
$831$	0	0
$832$	−3.56155	−0.123475
$833$	−20.2462	−0.701490
$834$	0	0
$835$	−50.7386	−1.75588
$836$	1.00000	0.0345857
$837$	0	0
$838$	23.1231	0.798774
$839$	−1.12311	−0.0387739	−0.0193870	−	0.999812i	$-0.506171\pi$
$839$	−1.12311	−0.0387739	−0.0193870	+	0.999812i	$0.506171\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	−1.06913	−0.0368447
$843$	0	0
$844$	−12.6847	−0.436624
$845$	0.630683	0.0216962
$846$	0	0
$847$	3.56155	0.122376
$848$	7.80776	0.268120
$849$	0	0
$850$	3.56155	0.122160
$851$	−17.3693	−0.595413
$852$	0	0
$853$	38.3542	1.31322	0.656611	−	0.754230i	$-0.271989\pi$
$853$	38.3542	1.31322	0.656611	+	0.754230i	$0.271989\pi$
$854$	−36.4924	−1.24874
$855$	0	0
$856$	−18.9309	−0.647044
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	−41.8617	−1.42830	−0.714152	−	0.699991i	$-0.753188\pi$
$859$	−41.8617	−1.42830	−0.714152	+	0.699991i	$0.753188\pi$
$860$	0	0
$861$	0	0
$862$	25.8617	0.880854
$863$	16.2462	0.553027	0.276514	−	0.961010i	$-0.410821\pi$
$863$	16.2462	0.553027	0.276514	+	0.961010i	$0.410821\pi$
$864$	0	0
$865$	36.0000	1.22404
$866$	31.3693	1.06597
$867$	0	0
$868$	7.12311	0.241774
$869$	4.00000	0.135691
$870$	0	0
$871$	−16.6847	−0.565338
$872$	−18.6847	−0.632742
$873$	0	0
$874$	5.56155	0.188122
$875$	42.7386	1.44483
$876$	0	0
$877$	−40.4384	−1.36551	−0.682755	−	0.730648i	$-0.739218\pi$
$877$	−40.4384	−1.36551	−0.682755	+	0.730648i	$0.739218\pi$
$878$	31.6155	1.06697
$879$	0	0
$880$	2.00000	0.0674200
$881$	28.7386	0.968229	0.484115	−	0.875005i	$-0.339142\pi$
$881$	28.7386	0.968229	0.484115	+	0.875005i	$0.339142\pi$
$882$	0	0
$883$	−54.7386	−1.84210	−0.921051	−	0.389442i	$-0.872668\pi$
$883$	−54.7386	−1.84210	−0.921051	+	0.389442i	$0.872668\pi$
$884$	12.6847	0.426631
$885$	0	0
$886$	17.8617	0.600077
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	−50.7386	−1.70172
$890$	−18.2462	−0.611614
$891$	0	0
$892$	−24.7386	−0.828311
$893$	8.00000	0.267710
$894$	0	0
$895$	44.4924	1.48722
$896$	3.56155	0.118983
$897$	0	0
$898$	17.6155	0.587838
$899$	−13.3693	−0.445892
$900$	0	0
$901$	−27.8078	−0.926411
$902$	2.00000	0.0665927
$903$	0	0
$904$	−5.12311	−0.170392
$905$	38.2462	1.27135
$906$	0	0
$907$	56.7926	1.88577	0.942884	−	0.333122i	$-0.108102\pi$
$907$	56.7926	1.88577	0.942884	+	0.333122i	$0.108102\pi$
$908$	17.1771	0.570041
$909$	0	0
$910$	25.3693	0.840985
$911$	46.9848	1.55668	0.778339	−	0.627845i	$-0.216063\pi$
$911$	46.9848	1.55668	0.778339	+	0.627845i	$0.216063\pi$
$912$	0	0
$913$	−2.24621	−0.0743387
$914$	32.4384	1.07297
$915$	0	0
$916$	21.1231	0.697927
$917$	25.3693	0.837769
$918$	0	0
$919$	28.4384	0.938098	0.469049	−	0.883172i	$-0.344597\pi$
$919$	28.4384	0.938098	0.469049	+	0.883172i	$0.344597\pi$
$920$	11.1231	0.366718
$921$	0	0
$922$	22.7386	0.748857
$923$	21.3693	0.703380
$924$	0	0
$925$	−3.12311	−0.102687
$926$	36.4924	1.19922
$927$	0	0
$928$	−6.68466	−0.219435
$929$	16.9309	0.555484	0.277742	−	0.960656i	$-0.410414\pi$
$929$	16.9309	0.555484	0.277742	+	0.960656i	$0.410414\pi$
$930$	0	0
$931$	−5.68466	−0.186307
$932$	−20.7386	−0.679317
$933$	0	0
$934$	−26.2462	−0.858802
$935$	−7.12311	−0.232950
$936$	0	0
$937$	−54.3002	−1.77391	−0.886955	−	0.461856i	$-0.847184\pi$
$937$	−54.3002	−1.77391	−0.886955	+	0.461856i	$0.847184\pi$
$938$	16.6847	0.544773
$939$	0	0
$940$	16.0000	0.521862
$941$	−20.4384	−0.666274	−0.333137	−	0.942878i	$-0.608107\pi$
$941$	−20.4384	−0.666274	−0.333137	+	0.942878i	$0.608107\pi$
$942$	0	0
$943$	11.1231	0.362218
$944$	−4.68466	−0.152473
$945$	0	0
$946$	0	0
$947$	−28.9848	−0.941881	−0.470940	−	0.882165i	$-0.656085\pi$
$947$	−28.9848	−0.941881	−0.470940	+	0.882165i	$0.656085\pi$
$948$	0	0
$949$	−9.56155	−0.310381
$950$	1.00000	0.0324443
$951$	0	0
$952$	−12.6847	−0.411112
$953$	9.12311	0.295526	0.147763	−	0.989023i	$-0.452793\pi$
$953$	9.12311	0.295526	0.147763	+	0.989023i	$0.452793\pi$
$954$	0	0
$955$	−41.3693	−1.33868
$956$	4.93087	0.159476
$957$	0	0
$958$	−11.3693	−0.367326
$959$	−30.0540	−0.970493
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−11.1231	−0.358623
$963$	0	0
$964$	21.6155	0.696189
$965$	2.24621	0.0723081
$966$	0	0
$967$	−3.86174	−0.124185	−0.0620926	−	0.998070i	$-0.519777\pi$
$967$	−3.86174	−0.124185	−0.0620926	+	0.998070i	$0.519777\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−2.24621	−0.0721215
$971$	14.2462	0.457183	0.228591	−	0.973522i	$-0.426588\pi$
$971$	14.2462	0.457183	0.228591	+	0.973522i	$0.426588\pi$
$972$	0	0
$973$	−61.8617	−1.98320
$974$	22.8769	0.733023
$975$	0	0
$976$	−10.2462	−0.327973
$977$	−55.3693	−1.77142	−0.885711	−	0.464238i	$-0.846328\pi$
$977$	−55.3693	−1.77142	−0.885711	+	0.464238i	$0.846328\pi$
$978$	0	0
$979$	−9.12311	−0.291576
$980$	−11.3693	−0.363180
$981$	0	0
$982$	14.6307	0.466884
$983$	48.2462	1.53882	0.769408	−	0.638758i	$-0.220552\pi$
$983$	48.2462	1.53882	0.769408	+	0.638758i	$0.220552\pi$
$984$	0	0
$985$	3.50758	0.111761
$986$	23.8078	0.758194
$987$	0	0
$988$	3.56155	0.113308
$989$	0	0
$990$	0	0
$991$	21.1231	0.670998	0.335499	−	0.942041i	$-0.391095\pi$
$991$	21.1231	0.670998	0.335499	+	0.942041i	$0.391095\pi$
$992$	2.00000	0.0635001
$993$	0	0
$994$	−21.3693	−0.677794
$995$	1.36932	0.0434103
$996$	0	0
$997$	−53.3693	−1.69022	−0.845112	−	0.534590i	$-0.820466\pi$
$997$	−53.3693	−1.69022	−0.845112	+	0.534590i	$0.820466\pi$
$998$	26.2462	0.830809
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3762.2.a.y.1.2		2
3.2	odd	2		418.2.a.e.1.1	✓	2
12.11	even	2		3344.2.a.k.1.2		2
33.32	even	2		4598.2.a.bj.1.1		2
57.56	even	2		7942.2.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.e.1.1	✓	2	3.2	odd	2
3344.2.a.k.1.2		2	12.11	even	2
3762.2.a.y.1.2		2	1.1	even	1	trivial
4598.2.a.bj.1.1		2	33.32	even	2
7942.2.a.x.1.2		2	57.56	even	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:	\( N \)	\(=\)	\( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3762.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +3.56155 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +3.56155 q^{7} +1.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} -3.56155 q^{13} +3.56155 q^{14} +1.00000 q^{16} -3.56155 q^{17} -1.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} -5.56155 q^{23} -1.00000 q^{25} -3.56155 q^{26} +3.56155 q^{28} -6.68466 q^{29} +2.00000 q^{31} +1.00000 q^{32} -3.56155 q^{34} -7.12311 q^{35} +3.12311 q^{37} -1.00000 q^{38} -2.00000 q^{40} -2.00000 q^{41} -1.00000 q^{44} -5.56155 q^{46} -8.00000 q^{47} +5.68466 q^{49} -1.00000 q^{50} -3.56155 q^{52} +7.80776 q^{53} +2.00000 q^{55} +3.56155 q^{56} -6.68466 q^{58} -4.68466 q^{59} -10.2462 q^{61} +2.00000 q^{62} +1.00000 q^{64} +7.12311 q^{65} +4.68466 q^{67} -3.56155 q^{68} -7.12311 q^{70} -6.00000 q^{71} +2.68466 q^{73} +3.12311 q^{74} -1.00000 q^{76} -3.56155 q^{77} -4.00000 q^{79} -2.00000 q^{80} -2.00000 q^{82} +2.24621 q^{83} +7.12311 q^{85} -1.00000 q^{88} +9.12311 q^{89} -12.6847 q^{91} -5.56155 q^{92} -8.00000 q^{94} +2.00000 q^{95} +1.12311 q^{97} +5.68466 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 3 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8} - 4 q^{10} - 2 q^{11} - 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} - 2 q^{19} - 4 q^{20} - 2 q^{22} - 7 q^{23} - 2 q^{25} - 3 q^{26} + 3 q^{28} - q^{29} + 4 q^{31} + 2 q^{32} - 3 q^{34} - 6 q^{35} - 2 q^{37} - 2 q^{38} - 4 q^{40} - 4 q^{41} - 2 q^{44} - 7 q^{46} - 16 q^{47} - q^{49} - 2 q^{50} - 3 q^{52} - 5 q^{53} + 4 q^{55} + 3 q^{56} - q^{58} + 3 q^{59} - 4 q^{61} + 4 q^{62} + 2 q^{64} + 6 q^{65} - 3 q^{67} - 3 q^{68} - 6 q^{70} - 12 q^{71} - 7 q^{73} - 2 q^{74} - 2 q^{76} - 3 q^{77} - 8 q^{79} - 4 q^{80} - 4 q^{82} - 12 q^{83} + 6 q^{85} - 2 q^{88} + 10 q^{89} - 13 q^{91} - 7 q^{92} - 16 q^{94} + 4 q^{95} - 6 q^{97} - q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 3 * q^7 + 2 * q^8 - 4 * q^10 - 2 * q^11 - 3 * q^13 + 3 * q^14 + 2 * q^16 - 3 * q^17 - 2 * q^19 - 4 * q^20 - 2 * q^22 - 7 * q^23 - 2 * q^25 - 3 * q^26 + 3 * q^28 - q^29 + 4 * q^31 + 2 * q^32 - 3 * q^34 - 6 * q^35 - 2 * q^37 - 2 * q^38 - 4 * q^40 - 4 * q^41 - 2 * q^44 - 7 * q^46 - 16 * q^47 - q^49 - 2 * q^50 - 3 * q^52 - 5 * q^53 + 4 * q^55 + 3 * q^56 - q^58 + 3 * q^59 - 4 * q^61 + 4 * q^62 + 2 * q^64 + 6 * q^65 - 3 * q^67 - 3 * q^68 - 6 * q^70 - 12 * q^71 - 7 * q^73 - 2 * q^74 - 2 * q^76 - 3 * q^77 - 8 * q^79 - 4 * q^80 - 4 * q^82 - 12 * q^83 + 6 * q^85 - 2 * q^88 + 10 * q^89 - 13 * q^91 - 7 * q^92 - 16 * q^94 + 4 * q^95 - 6 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more