LMFDB - Embedded newform 8037.2.a.o.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 28 x^{16} + 25 x^{15} + 322 x^{14} - 247 x^{13} - 1971 x^{12} + 1231 x^{11} + 6953 x^{10} + \cdots + 9 $ x^18 - x^17 - 28x^16 + 25x^15 + 322x^14 - 247x^13 - 1971x^12 + 1231x^11 + 6953x^10 - 3283x^9 - 14235x^8 + 4562x^7 + 15962x^6 - 2882x^5 - 8159x^4 + 606x^3 + 890x^2 - 179x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-1.40846$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.40846 q^{2} -0.0162457 q^{4} -0.976908 q^{5} -4.66797 q^{7} -2.83980 q^{8} +O(q^{10})$	\(q+1.40846 q^{2} -0.0162457 q^{4} -0.976908 q^{5} -4.66797 q^{7} -2.83980 q^{8} -1.37593 q^{10} +0.133750 q^{11} -4.60501 q^{13} -6.57464 q^{14} -3.96724 q^{16} -0.535675 q^{17} -1.00000 q^{19} +0.0158705 q^{20} +0.188381 q^{22} -6.18211 q^{23} -4.04565 q^{25} -6.48597 q^{26} +0.0758342 q^{28} +6.83121 q^{29} -5.08467 q^{31} +0.0918970 q^{32} -0.754475 q^{34} +4.56017 q^{35} -4.65581 q^{37} -1.40846 q^{38} +2.77422 q^{40} +0.799649 q^{41} +5.28565 q^{43} -0.00217286 q^{44} -8.70724 q^{46} -1.00000 q^{47} +14.7899 q^{49} -5.69813 q^{50} +0.0748115 q^{52} -10.5557 q^{53} -0.130661 q^{55} +13.2561 q^{56} +9.62147 q^{58} -13.5870 q^{59} -12.9782 q^{61} -7.16154 q^{62} +8.06392 q^{64} +4.49867 q^{65} +13.2617 q^{67} +0.00870239 q^{68} +6.42282 q^{70} -3.59575 q^{71} +8.11956 q^{73} -6.55751 q^{74} +0.0162457 q^{76} -0.624340 q^{77} -12.4922 q^{79} +3.87563 q^{80} +1.12627 q^{82} +1.21594 q^{83} +0.523305 q^{85} +7.44462 q^{86} -0.379823 q^{88} -6.48351 q^{89} +21.4960 q^{91} +0.100432 q^{92} -1.40846 q^{94} +0.976908 q^{95} +8.87520 q^{97} +20.8310 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * 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 $

$2$	1.40846	0.995930	0.497965	−	0.867197i	$-0.334081\pi$
$2$	1.40846	0.995930	0.497965	+	0.867197i	$0.334081\pi$
$3$	0	0
$3$	0	0
$4$	−0.0162457	−0.00812283
$5$	−0.976908	−0.436887	−0.218443	−	0.975850i	$-0.570098\pi$
$5$	−0.976908	−0.436887	−0.218443	+	0.975850i	$0.570098\pi$
$6$	0	0
$7$	−4.66797	−1.76433	−0.882163	−	0.470945i	$-0.843913\pi$
$7$	−4.66797	−1.76433	−0.882163	+	0.470945i	$0.843913\pi$
$8$	−2.83980	−1.00402
$9$	0	0
$10$	−1.37593	−0.435109
$11$	0.133750	0.0403271	0.0201636	−	0.999797i	$-0.493581\pi$
$11$	0.133750	0.0403271	0.0201636	+	0.999797i	$0.493581\pi$
$12$	0	0
$13$	−4.60501	−1.27720	−0.638600	−	0.769538i	$-0.720486\pi$
$13$	−4.60501	−1.27720	−0.638600	+	0.769538i	$0.720486\pi$
$14$	−6.57464	−1.75715
$15$	0	0
$16$	−3.96724	−0.991811
$17$	−0.535675	−0.129920	−0.0649601	−	0.997888i	$-0.520692\pi$
$17$	−0.535675	−0.129920	−0.0649601	+	0.997888i	$0.520692\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.0158705	0.00354876
$21$	0	0
$22$	0.188381	0.0401630
$23$	−6.18211	−1.28906	−0.644530	−	0.764579i	$-0.722947\pi$
$23$	−6.18211	−1.28906	−0.644530	+	0.764579i	$0.722947\pi$
$24$	0	0
$25$	−4.04565	−0.809130
$26$	−6.48597	−1.27200
$27$	0	0
$28$	0.0758342	0.0143313
$29$	6.83121	1.26852	0.634262	−	0.773118i	$-0.281304\pi$
$29$	6.83121	1.26852	0.634262	+	0.773118i	$0.281304\pi$
$30$	0	0
$31$	−5.08467	−0.913233	−0.456617	−	0.889664i	$-0.650939\pi$
$31$	−5.08467	−0.913233	−0.456617	+	0.889664i	$0.650939\pi$
$32$	0.0918970	0.0162453
$33$	0	0
$34$	−0.754475	−0.129391
$35$	4.56017	0.770810
$36$	0	0
$37$	−4.65581	−0.765410	−0.382705	−	0.923871i	$-0.625007\pi$
$37$	−4.65581	−0.765410	−0.382705	+	0.923871i	$0.625007\pi$
$38$	−1.40846	−0.228482
$39$	0	0
$40$	2.77422	0.438643
$41$	0.799649	0.124884	0.0624421	−	0.998049i	$-0.480111\pi$
$41$	0.799649	0.124884	0.0624421	+	0.998049i	$0.480111\pi$
$42$	0	0
$43$	5.28565	0.806055	0.403027	−	0.915188i	$-0.367958\pi$
$43$	5.28565	0.806055	0.403027	+	0.915188i	$0.367958\pi$
$44$	−0.00217286	−0.000327570 0
$45$	0	0
$46$	−8.70724	−1.28381
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	14.7899	2.11284
$50$	−5.69813	−0.805837
$51$	0	0
$52$	0.0748115	0.0103745
$53$	−10.5557	−1.44994	−0.724971	−	0.688780i	$-0.758147\pi$
$53$	−10.5557	−1.44994	−0.724971	+	0.688780i	$0.758147\pi$
$54$	0	0
$55$	−0.130661	−0.0176184
$56$	13.2561	1.77142
$57$	0	0
$58$	9.62147	1.26336
$59$	−13.5870	−1.76887	−0.884437	−	0.466659i	$-0.845458\pi$
$59$	−13.5870	−1.76887	−0.884437	+	0.466659i	$0.845458\pi$
$60$	0	0
$61$	−12.9782	−1.66168	−0.830842	−	0.556508i	$-0.812141\pi$
$61$	−12.9782	−1.66168	−0.830842	+	0.556508i	$0.812141\pi$
$62$	−7.16154	−0.909517
$63$	0	0
$64$	8.06392	1.00799
$65$	4.49867	0.557992
$66$	0	0
$67$	13.2617	1.62017	0.810085	−	0.586312i	$-0.199421\pi$
$67$	13.2617	1.62017	0.810085	+	0.586312i	$0.199421\pi$
$68$	0.00870239	0.00105532
$69$	0	0
$70$	6.42282	0.767673
$71$	−3.59575	−0.426737	−0.213368	−	0.976972i	$-0.568443\pi$
$71$	−3.59575	−0.426737	−0.213368	+	0.976972i	$0.568443\pi$
$72$	0	0
$73$	8.11956	0.950323	0.475161	−	0.879899i	$-0.342390\pi$
$73$	8.11956	0.950323	0.475161	+	0.879899i	$0.342390\pi$
$74$	−6.55751	−0.762295
$75$	0	0
$76$	0.0162457	0.00186350
$77$	−0.624340	−0.0711502
$78$	0	0
$79$	−12.4922	−1.40549	−0.702743	−	0.711444i	$-0.748041\pi$
$79$	−12.4922	−1.40549	−0.702743	+	0.711444i	$0.748041\pi$
$80$	3.87563	0.433309
$81$	0	0
$82$	1.12627	0.124376
$83$	1.21594	0.133467	0.0667336	−	0.997771i	$-0.478742\pi$
$83$	1.21594	0.133467	0.0667336	+	0.997771i	$0.478742\pi$
$84$	0	0
$85$	0.523305	0.0567604
$86$	7.44462	0.802774
$87$	0	0
$88$	−0.379823	−0.0404892
$89$	−6.48351	−0.687251	−0.343625	−	0.939107i	$-0.611655\pi$
$89$	−6.48351	−0.687251	−0.343625	+	0.939107i	$0.611655\pi$
$90$	0	0
$91$	21.4960	2.25340
$92$	0.100432	0.0104708
$93$	0	0
$94$	−1.40846	−0.145271
$95$	0.976908	0.100229
$96$	0	0
$97$	8.87520	0.901140	0.450570	−	0.892741i	$-0.351221\pi$
$97$	8.87520	0.901140	0.450570	+	0.892741i	$0.351221\pi$
$98$	20.8310	2.10425
$99$	0	0
$100$	0.0657243	0.00657243
$101$	−5.31592	−0.528954	−0.264477	−	0.964392i	$-0.585199\pi$
$101$	−5.31592	−0.528954	−0.264477	+	0.964392i	$0.585199\pi$
$102$	0	0
$103$	16.0383	1.58030	0.790150	−	0.612914i	$-0.210003\pi$
$103$	16.0383	1.58030	0.790150	+	0.612914i	$0.210003\pi$
$104$	13.0773	1.28234
$105$	0	0
$106$	−14.8673	−1.44404
$107$	−2.56956	−0.248409	−0.124205	−	0.992257i	$-0.539638\pi$
$107$	−2.56956	−0.248409	−0.124205	+	0.992257i	$0.539638\pi$
$108$	0	0
$109$	5.72029	0.547904	0.273952	−	0.961743i	$-0.411669\pi$
$109$	5.72029	0.547904	0.273952	+	0.961743i	$0.411669\pi$
$110$	−0.184031	−0.0175467
$111$	0	0
$112$	18.5190	1.74988
$113$	14.1137	1.32770	0.663852	−	0.747864i	$-0.268920\pi$
$113$	14.1137	1.32770	0.663852	+	0.747864i	$0.268920\pi$
$114$	0	0
$115$	6.03935	0.563173
$116$	−0.110977	−0.0103040
$117$	0	0
$118$	−19.1367	−1.76168
$119$	2.50051	0.229222
$120$	0	0
$121$	−10.9821	−0.998374
$122$	−18.2792	−1.65492
$123$	0	0
$124$	0.0826038	0.00741804
$125$	8.83677	0.790385
$126$	0	0
$127$	15.7217	1.39507	0.697536	−	0.716550i	$-0.254280\pi$
$127$	15.7217	1.39507	0.697536	+	0.716550i	$0.254280\pi$
$128$	11.1739	0.987643
$129$	0	0
$130$	6.33620	0.555721
$131$	13.4414	1.17438	0.587191	−	0.809449i	$-0.300234\pi$
$131$	13.4414	1.17438	0.587191	+	0.809449i	$0.300234\pi$
$132$	0	0
$133$	4.66797	0.404764
$134$	18.6785	1.61358
$135$	0	0
$136$	1.52121	0.130443
$137$	−6.23823	−0.532968	−0.266484	−	0.963839i	$-0.585862\pi$
$137$	−6.23823	−0.532968	−0.266484	+	0.963839i	$0.585862\pi$
$138$	0	0
$139$	20.8630	1.76958	0.884790	−	0.465990i	$-0.154302\pi$
$139$	20.8630	1.76958	0.884790	+	0.465990i	$0.154302\pi$
$140$	−0.0740830	−0.00626116
$141$	0	0
$142$	−5.06446	−0.425000
$143$	−0.615920	−0.0515058
$144$	0	0
$145$	−6.67346	−0.554201
$146$	11.4361	0.946455
$147$	0	0
$148$	0.0756366	0.00621729
$149$	11.9648	0.980195	0.490097	−	0.871668i	$-0.336961\pi$
$149$	11.9648	0.980195	0.490097	+	0.871668i	$0.336961\pi$
$150$	0	0
$151$	−6.35684	−0.517312	−0.258656	−	0.965970i	$-0.583280\pi$
$151$	−6.35684	−0.517312	−0.258656	+	0.965970i	$0.583280\pi$
$152$	2.83980	0.230338
$153$	0	0
$154$	−0.879357	−0.0708606
$155$	4.96725	0.398979
$156$	0	0
$157$	−20.6372	−1.64703	−0.823515	−	0.567295i	$-0.807990\pi$
$157$	−20.6372	−1.64703	−0.823515	+	0.567295i	$0.807990\pi$
$158$	−17.5948	−1.39977
$159$	0	0
$160$	−0.0897750	−0.00709733
$161$	28.8579	2.27432
$162$	0	0
$163$	−20.8194	−1.63070	−0.815351	−	0.578967i	$-0.803456\pi$
$163$	−20.8194	−1.63070	−0.815351	+	0.578967i	$0.803456\pi$
$164$	−0.0129908	−0.00101441
$165$	0	0
$166$	1.71261	0.132924
$167$	10.8417	0.838956	0.419478	−	0.907766i	$-0.362213\pi$
$167$	10.8417	0.838956	0.419478	+	0.907766i	$0.362213\pi$
$168$	0	0
$169$	8.20615	0.631242
$170$	0.737053	0.0565294
$171$	0	0
$172$	−0.0858689	−0.00654744
$173$	−17.8205	−1.35486	−0.677432	−	0.735585i	$-0.736907\pi$
$173$	−17.8205	−1.35486	−0.677432	+	0.735585i	$0.736907\pi$
$174$	0	0
$175$	18.8850	1.42757
$176$	−0.530619	−0.0399969
$177$	0	0
$178$	−9.13175	−0.684454
$179$	−11.8946	−0.889046	−0.444523	−	0.895768i	$-0.646627\pi$
$179$	−11.8946	−0.889046	−0.444523	+	0.895768i	$0.646627\pi$
$180$	0	0
$181$	11.8433	0.880305	0.440152	−	0.897923i	$-0.354924\pi$
$181$	11.8433	0.880305	0.440152	+	0.897923i	$0.354924\pi$
$182$	30.2763	2.24423
$183$	0	0
$184$	17.5559	1.29424
$185$	4.54830	0.334397
$186$	0	0
$187$	−0.0716465	−0.00523931
$188$	0.0162457	0.00118484
$189$	0	0
$190$	1.37593	0.0998208
$191$	−14.9050	−1.07849	−0.539243	−	0.842150i	$-0.681290\pi$
$191$	−14.9050	−1.07849	−0.539243	+	0.842150i	$0.681290\pi$
$192$	0	0
$193$	5.54493	0.399133	0.199566	−	0.979884i	$-0.436047\pi$
$193$	5.54493	0.399133	0.199566	+	0.979884i	$0.436047\pi$
$194$	12.5004	0.897473
$195$	0	0
$196$	−0.240272	−0.0171623
$197$	17.2938	1.23213	0.616066	−	0.787694i	$-0.288725\pi$
$197$	17.2938	1.23213	0.616066	+	0.787694i	$0.288725\pi$
$198$	0	0
$199$	−13.3102	−0.943533	−0.471766	−	0.881724i	$-0.656383\pi$
$199$	−13.3102	−0.943533	−0.471766	+	0.881724i	$0.656383\pi$
$200$	11.4888	0.812383
$201$	0	0
$202$	−7.48725	−0.526801
$203$	−31.8878	−2.23809
$204$	0	0
$205$	−0.781184	−0.0545603
$206$	22.5893	1.57387
$207$	0	0
$208$	18.2692	1.26674
$209$	−0.133750	−0.00925168
$210$	0	0
$211$	−3.61791	−0.249067	−0.124534	−	0.992215i	$-0.539743\pi$
$211$	−3.61791	−0.249067	−0.124534	+	0.992215i	$0.539743\pi$
$212$	0.171485	0.0117776
$213$	0	0
$214$	−3.61912	−0.247398
$215$	−5.16360	−0.352154
$216$	0	0
$217$	23.7351	1.61124
$218$	8.05679	0.545675
$219$	0	0
$220$	0.00212268	0.000143111 0
$221$	2.46679	0.165934
$222$	0	0
$223$	−2.69806	−0.180675	−0.0903377	−	0.995911i	$-0.528795\pi$
$223$	−2.69806	−0.180675	−0.0903377	+	0.995911i	$0.528795\pi$
$224$	−0.428972	−0.0286619
$225$	0	0
$226$	19.8785	1.32230
$227$	18.5532	1.23142	0.615708	−	0.787974i	$-0.288870\pi$
$227$	18.5532	1.23142	0.615708	+	0.787974i	$0.288870\pi$
$228$	0	0
$229$	−6.18564	−0.408759	−0.204379	−	0.978892i	$-0.565518\pi$
$229$	−6.18564	−0.408759	−0.204379	+	0.978892i	$0.565518\pi$
$230$	8.50618	0.560881
$231$	0	0
$232$	−19.3992	−1.27362
$233$	−13.6729	−0.895742	−0.447871	−	0.894098i	$-0.647818\pi$
$233$	−13.6729	−0.895742	−0.447871	+	0.894098i	$0.647818\pi$
$234$	0	0
$235$	0.976908	0.0637265
$236$	0.220729	0.0143683
$237$	0	0
$238$	3.52187	0.228289
$239$	−1.18228	−0.0764755	−0.0382377	−	0.999269i	$-0.512174\pi$
$239$	−1.18228	−0.0764755	−0.0382377	+	0.999269i	$0.512174\pi$
$240$	0	0
$241$	−21.9924	−1.41665	−0.708326	−	0.705885i	$-0.750550\pi$
$241$	−21.9924	−1.41665	−0.708326	+	0.705885i	$0.750550\pi$
$242$	−15.4678	−0.994311
$243$	0	0
$244$	0.210839	0.0134976
$245$	−14.4484	−0.923074
$246$	0	0
$247$	4.60501	0.293010
$248$	14.4394	0.916905
$249$	0	0
$250$	12.4462	0.787168
$251$	21.6132	1.36422	0.682108	−	0.731252i	$-0.261064\pi$
$251$	21.6132	1.36422	0.682108	+	0.731252i	$0.261064\pi$
$252$	0	0
$253$	−0.826857	−0.0519840
$254$	22.1433	1.38939
$255$	0	0
$256$	−0.389870	−0.0243669
$257$	1.43518	0.0895244	0.0447622	−	0.998998i	$-0.485747\pi$
$257$	1.43518	0.0895244	0.0447622	+	0.998998i	$0.485747\pi$
$258$	0	0
$259$	21.7332	1.35043
$260$	−0.0730839	−0.00453247
$261$	0	0
$262$	18.9317	1.16960
$263$	−14.3050	−0.882084	−0.441042	−	0.897486i	$-0.645391\pi$
$263$	−14.3050	−0.882084	−0.441042	+	0.897486i	$0.645391\pi$
$264$	0	0
$265$	10.3120	0.633460
$266$	6.57464	0.403117
$267$	0	0
$268$	−0.215444	−0.0131604
$269$	−14.9977	−0.914423	−0.457211	−	0.889358i	$-0.651152\pi$
$269$	−14.9977	−0.914423	−0.457211	+	0.889358i	$0.651152\pi$
$270$	0	0
$271$	14.8953	0.904827	0.452414	−	0.891808i	$-0.350563\pi$
$271$	14.8953	0.904827	0.452414	+	0.891808i	$0.350563\pi$
$272$	2.12515	0.128856
$273$	0	0
$274$	−8.78629	−0.530799
$275$	−0.541106	−0.0326299
$276$	0	0
$277$	−20.4168	−1.22673	−0.613365	−	0.789800i	$-0.710184\pi$
$277$	−20.4168	−1.22673	−0.613365	+	0.789800i	$0.710184\pi$
$278$	29.3847	1.76238
$279$	0	0
$280$	−12.9500	−0.773909
$281$	−30.4220	−1.81482	−0.907411	−	0.420244i	$-0.861944\pi$
$281$	−30.4220	−1.81482	−0.907411	+	0.420244i	$0.861944\pi$
$282$	0	0
$283$	31.9242	1.89770	0.948849	−	0.315730i	$-0.102250\pi$
$283$	31.9242	1.89770	0.948849	+	0.315730i	$0.102250\pi$
$284$	0.0584153	0.00346631
$285$	0	0
$286$	−0.867498	−0.0512962
$287$	−3.73274	−0.220336
$288$	0	0
$289$	−16.7131	−0.983121
$290$	−9.39929	−0.551945
$291$	0	0
$292$	−0.131908	−0.00771931
$293$	8.62521	0.503890	0.251945	−	0.967742i	$-0.418930\pi$
$293$	8.62521	0.503890	0.251945	+	0.967742i	$0.418930\pi$
$294$	0	0
$295$	13.2732	0.772798
$296$	13.2215	0.768487
$297$	0	0
$298$	16.8519	0.976206
$299$	28.4687	1.64639
$300$	0	0
$301$	−24.6733	−1.42214
$302$	−8.95334	−0.515207
$303$	0	0
$304$	3.96724	0.227537
$305$	12.6785	0.725968
$306$	0	0
$307$	9.21657	0.526017	0.263009	−	0.964793i	$-0.415285\pi$
$307$	9.21657	0.526017	0.263009	+	0.964793i	$0.415285\pi$
$308$	0.0101428	0.000577941 0
$309$	0	0
$310$	6.99617	0.397356
$311$	0.584881	0.0331656	0.0165828	−	0.999862i	$-0.494721\pi$
$311$	0.584881	0.0331656	0.0165828	+	0.999862i	$0.494721\pi$
$312$	0	0
$313$	22.2332	1.25669	0.628347	−	0.777933i	$-0.283732\pi$
$313$	22.2332	1.25669	0.628347	+	0.777933i	$0.283732\pi$
$314$	−29.0667	−1.64033
$315$	0	0
$316$	0.202944	0.0114165
$317$	−3.00849	−0.168974	−0.0844869	−	0.996425i	$-0.526925\pi$
$317$	−3.00849	−0.168974	−0.0844869	+	0.996425i	$0.526925\pi$
$318$	0	0
$319$	0.913674	0.0511559
$320$	−7.87771	−0.440377
$321$	0	0
$322$	40.6451	2.26506
$323$	0.535675	0.0298057
$324$	0	0
$325$	18.6303	1.03342
$326$	−29.3233	−1.62407
$327$	0	0
$328$	−2.27084	−0.125386
$329$	4.66797	0.257353
$330$	0	0
$331$	−18.7739	−1.03191	−0.515953	−	0.856617i	$-0.672562\pi$
$331$	−18.7739	−1.03191	−0.515953	+	0.856617i	$0.672562\pi$
$332$	−0.0197538	−0.00108413
$333$	0	0
$334$	15.2701	0.835541
$335$	−12.9554	−0.707831
$336$	0	0
$337$	23.5998	1.28556	0.642782	−	0.766049i	$-0.277780\pi$
$337$	23.5998	1.28556	0.642782	+	0.766049i	$0.277780\pi$
$338$	11.5580	0.628673
$339$	0	0
$340$	−0.00850143	−0.000461055 0
$341$	−0.680074	−0.0368281
$342$	0	0
$343$	−36.3631	−1.96342
$344$	−15.0102	−0.809295
$345$	0	0
$346$	−25.0994	−1.34935
$347$	−14.7985	−0.794426	−0.397213	−	0.917726i	$-0.630023\pi$
$347$	−14.7985	−0.794426	−0.397213	+	0.917726i	$0.630023\pi$
$348$	0	0
$349$	29.6330	1.58622	0.793108	−	0.609081i	$-0.208462\pi$
$349$	29.6330	1.58622	0.793108	+	0.609081i	$0.208462\pi$
$350$	26.5987	1.42176
$351$	0	0
$352$	0.0122912	0.000655124 0
$353$	−5.44740	−0.289936	−0.144968	−	0.989436i	$-0.546308\pi$
$353$	−5.44740	−0.289936	−0.144968	+	0.989436i	$0.546308\pi$
$354$	0	0
$355$	3.51271	0.186435
$356$	0.105329	0.00558242
$357$	0	0
$358$	−16.7531	−0.885427
$359$	30.9695	1.63451	0.817255	−	0.576276i	$-0.195495\pi$
$359$	30.9695	1.63451	0.817255	+	0.576276i	$0.195495\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	16.6808	0.876722
$363$	0	0
$364$	−0.349217	−0.0183040
$365$	−7.93207	−0.415183
$366$	0	0
$367$	−11.0009	−0.574240	−0.287120	−	0.957895i	$-0.592698\pi$
$367$	−11.0009	−0.574240	−0.287120	+	0.957895i	$0.592698\pi$
$368$	24.5259	1.27850
$369$	0	0
$370$	6.40608	0.333036
$371$	49.2738	2.55817
$372$	0	0
$373$	−30.1607	−1.56166	−0.780831	−	0.624742i	$-0.785204\pi$
$373$	−30.1607	−1.56166	−0.780831	+	0.624742i	$0.785204\pi$
$374$	−0.100911	−0.00521799
$375$	0	0
$376$	2.83980	0.146451
$377$	−31.4578	−1.62016
$378$	0	0
$379$	6.53436	0.335648	0.167824	−	0.985817i	$-0.446326\pi$
$379$	6.53436	0.335648	0.167824	+	0.985817i	$0.446326\pi$
$380$	−0.0158705	−0.000814140 0
$381$	0	0
$382$	−20.9930	−1.07410
$383$	−28.0827	−1.43496	−0.717478	−	0.696581i	$-0.754704\pi$
$383$	−28.0827	−1.43496	−0.717478	+	0.696581i	$0.754704\pi$
$384$	0	0
$385$	0.609923	0.0310846
$386$	7.80980	0.397508
$387$	0	0
$388$	−0.144184	−0.00731981
$389$	3.16103	0.160271	0.0801353	−	0.996784i	$-0.474465\pi$
$389$	3.16103	0.160271	0.0801353	+	0.996784i	$0.474465\pi$
$390$	0	0
$391$	3.31160	0.167475
$392$	−42.0004	−2.12134
$393$	0	0
$394$	24.3576	1.22712
$395$	12.2038	0.614038
$396$	0	0
$397$	−19.5535	−0.981361	−0.490681	−	0.871339i	$-0.663252\pi$
$397$	−19.5535	−0.981361	−0.490681	+	0.871339i	$0.663252\pi$
$398$	−18.7468	−0.939693
$399$	0	0
$400$	16.0501	0.802504
$401$	−6.05940	−0.302592	−0.151296	−	0.988488i	$-0.548345\pi$
$401$	−6.05940	−0.302592	−0.151296	+	0.988488i	$0.548345\pi$
$402$	0	0
$403$	23.4150	1.16638
$404$	0.0863606	0.00429660
$405$	0	0
$406$	−44.9127	−2.22898
$407$	−0.622714	−0.0308668
$408$	0	0
$409$	−15.5479	−0.768793	−0.384396	−	0.923168i	$-0.625590\pi$
$409$	−15.5479	−0.768793	−0.384396	+	0.923168i	$0.625590\pi$
$410$	−1.10026	−0.0543382
$411$	0	0
$412$	−0.260553	−0.0128365
$413$	63.4236	3.12087
$414$	0	0
$415$	−1.18787	−0.0583101
$416$	−0.423187	−0.0207485
$417$	0	0
$418$	−0.188381	−0.00921403
$419$	−23.4943	−1.14777	−0.573887	−	0.818935i	$-0.694565\pi$
$419$	−23.4943	−1.14777	−0.573887	+	0.818935i	$0.694565\pi$
$420$	0	0
$421$	36.2240	1.76545	0.882725	−	0.469890i	$-0.155707\pi$
$421$	36.2240	1.76545	0.882725	+	0.469890i	$0.155707\pi$
$422$	−5.09568	−0.248054
$423$	0	0
$424$	29.9761	1.45577
$425$	2.16715	0.105122
$426$	0	0
$427$	60.5817	2.93175
$428$	0.0417443	0.00201779
$429$	0	0
$430$	−7.27271	−0.350721
$431$	6.35000	0.305869	0.152934	−	0.988236i	$-0.451128\pi$
$431$	6.35000	0.305869	0.152934	+	0.988236i	$0.451128\pi$
$432$	0	0
$433$	13.6697	0.656926	0.328463	−	0.944517i	$-0.393469\pi$
$433$	13.6697	0.656926	0.328463	+	0.944517i	$0.393469\pi$
$434$	33.4298	1.60468
$435$	0	0
$436$	−0.0929299	−0.00445053
$437$	6.18211	0.295730
$438$	0	0
$439$	3.15985	0.150811	0.0754057	−	0.997153i	$-0.475975\pi$
$439$	3.15985	0.150811	0.0754057	+	0.997153i	$0.475975\pi$
$440$	0.371052	0.0176892
$441$	0	0
$442$	3.47437	0.165259
$443$	10.9632	0.520876	0.260438	−	0.965491i	$-0.416133\pi$
$443$	10.9632	0.520876	0.260438	+	0.965491i	$0.416133\pi$
$444$	0	0
$445$	6.33379	0.300251
$446$	−3.80011	−0.179940
$447$	0	0
$448$	−37.6421	−1.77842
$449$	−13.9890	−0.660182	−0.330091	−	0.943949i	$-0.607079\pi$
$449$	−13.9890	−0.660182	−0.330091	+	0.943949i	$0.607079\pi$
$450$	0	0
$451$	0.106953	0.00503622
$452$	−0.229286	−0.0107847
$453$	0	0
$454$	26.1314	1.22641
$455$	−20.9997	−0.984480
$456$	0	0
$457$	−18.6690	−0.873301	−0.436650	−	0.899631i	$-0.643835\pi$
$457$	−18.6690	−0.873301	−0.436650	+	0.899631i	$0.643835\pi$
$458$	−8.71221	−0.407095
$459$	0	0
$460$	−0.0981133	−0.00457456
$461$	−17.0972	−0.796297	−0.398148	−	0.917321i	$-0.630347\pi$
$461$	−17.0972	−0.796297	−0.398148	+	0.917321i	$0.630347\pi$
$462$	0	0
$463$	−6.33715	−0.294512	−0.147256	−	0.989098i	$-0.547044\pi$
$463$	−6.33715	−0.294512	−0.147256	+	0.989098i	$0.547044\pi$
$464$	−27.1011	−1.25814
$465$	0	0
$466$	−19.2577	−0.892096
$467$	−25.7393	−1.19107	−0.595535	−	0.803329i	$-0.703060\pi$
$467$	−25.7393	−1.19107	−0.595535	+	0.803329i	$0.703060\pi$
$468$	0	0
$469$	−61.9050	−2.85851
$470$	1.37593	0.0634671
$471$	0	0
$472$	38.5843	1.77599
$473$	0.706956	0.0325059
$474$	0	0
$475$	4.04565	0.185627
$476$	−0.0406225	−0.00186193
$477$	0	0
$478$	−1.66520	−0.0761643
$479$	−31.2547	−1.42806	−0.714032	−	0.700113i	$-0.753133\pi$
$479$	−31.2547	−1.42806	−0.714032	+	0.700113i	$0.753133\pi$
$480$	0	0
$481$	21.4401	0.977582
$482$	−30.9753	−1.41089
$483$	0	0
$484$	0.178412	0.00810962
$485$	−8.67026	−0.393696
$486$	0	0
$487$	20.1333	0.912325	0.456162	−	0.889897i	$-0.349224\pi$
$487$	20.1333	0.912325	0.456162	+	0.889897i	$0.349224\pi$
$488$	36.8554	1.66836
$489$	0	0
$490$	−20.3500	−0.919317
$491$	−17.9322	−0.809267	−0.404634	−	0.914479i	$-0.632601\pi$
$491$	−17.9322	−0.809267	−0.404634	+	0.914479i	$0.632601\pi$
$492$	0	0
$493$	−3.65931	−0.164807
$494$	6.48597	0.291818
$495$	0	0
$496$	20.1721	0.905755
$497$	16.7848	0.752902
$498$	0	0
$499$	4.55597	0.203953	0.101977	−	0.994787i	$-0.467483\pi$
$499$	4.55597	0.203953	0.101977	+	0.994787i	$0.467483\pi$
$500$	−0.143559	−0.00642016
$501$	0	0
$502$	30.4413	1.35866
$503$	36.0062	1.60544	0.802719	−	0.596357i	$-0.203386\pi$
$503$	36.0062	1.60544	0.802719	+	0.596357i	$0.203386\pi$
$504$	0	0
$505$	5.19316	0.231093
$506$	−1.16459	−0.0517725
$507$	0	0
$508$	−0.255409	−0.0113319
$509$	−37.4450	−1.65972	−0.829859	−	0.557973i	$-0.811579\pi$
$509$	−37.4450	−1.65972	−0.829859	+	0.557973i	$0.811579\pi$
$510$	0	0
$511$	−37.9018	−1.67668
$512$	−22.8969	−1.01191
$513$	0	0
$514$	2.02140	0.0891600
$515$	−15.6679	−0.690412
$516$	0	0
$517$	−0.133750	−0.00588232
$518$	30.6102	1.34494
$519$	0	0
$520$	−12.7753	−0.560235
$521$	−19.6554	−0.861119	−0.430560	−	0.902562i	$-0.641684\pi$
$521$	−19.6554	−0.861119	−0.430560	+	0.902562i	$0.641684\pi$
$522$	0	0
$523$	−2.64979	−0.115867	−0.0579337	−	0.998320i	$-0.518451\pi$
$523$	−2.64979	−0.115867	−0.0579337	+	0.998320i	$0.518451\pi$
$524$	−0.218365	−0.00953930
$525$	0	0
$526$	−20.1480	−0.878494
$527$	2.72373	0.118647
$528$	0	0
$529$	15.2185	0.661673
$530$	14.5240	0.630882
$531$	0	0
$532$	−0.0758342	−0.00328783
$533$	−3.68240	−0.159502
$534$	0	0
$535$	2.51023	0.108527
$536$	−37.6604	−1.62668
$537$	0	0
$538$	−21.1236	−0.910702
$539$	1.97815	0.0852050
$540$	0	0
$541$	−1.07632	−0.0462747	−0.0231373	−	0.999732i	$-0.507365\pi$
$541$	−1.07632	−0.0462747	−0.0231373	+	0.999732i	$0.507365\pi$
$542$	20.9795	0.901145
$543$	0	0
$544$	−0.0492269	−0.00211059
$545$	−5.58820	−0.239372
$546$	0	0
$547$	5.84699	0.249999	0.125000	−	0.992157i	$-0.460107\pi$
$547$	5.84699	0.249999	0.125000	+	0.992157i	$0.460107\pi$
$548$	0.101344	0.00432921
$549$	0	0
$550$	−0.762124	−0.0324971
$551$	−6.83121	−0.291019
$552$	0	0
$553$	58.3133	2.47973
$554$	−28.7563	−1.22174
$555$	0	0
$556$	−0.338934	−0.0143740
$557$	−18.7115	−0.792830	−0.396415	−	0.918071i	$-0.629746\pi$
$557$	−18.7115	−0.792830	−0.396415	+	0.918071i	$0.629746\pi$
$558$	0	0
$559$	−24.3405	−1.02949
$560$	−18.0913	−0.764498
$561$	0	0
$562$	−42.8481	−1.80744
$563$	34.6722	1.46126	0.730630	−	0.682773i	$-0.239226\pi$
$563$	34.6722	1.46126	0.730630	+	0.682773i	$0.239226\pi$
$564$	0	0
$565$	−13.7878	−0.580056
$566$	44.9639	1.88997
$567$	0	0
$568$	10.2112	0.428452
$569$	15.4242	0.646615	0.323307	−	0.946294i	$-0.395205\pi$
$569$	15.4242	0.646615	0.323307	+	0.946294i	$0.395205\pi$
$570$	0	0
$571$	−7.08566	−0.296526	−0.148263	−	0.988948i	$-0.547368\pi$
$571$	−7.08566	−0.296526	−0.148263	+	0.988948i	$0.547368\pi$
$572$	0.0100060	0.000418373 0
$573$	0	0
$574$	−5.25740	−0.219440
$575$	25.0107	1.04302
$576$	0	0
$577$	−4.14207	−0.172437	−0.0862183	−	0.996276i	$-0.527478\pi$
$577$	−4.14207	−0.172437	−0.0862183	+	0.996276i	$0.527478\pi$
$578$	−23.5396	−0.979120
$579$	0	0
$580$	0.108415	0.00450168
$581$	−5.67599	−0.235480
$582$	0	0
$583$	−1.41183	−0.0584720
$584$	−23.0579	−0.954143
$585$	0	0
$586$	12.1482	0.501839
$587$	−13.2078	−0.545144	−0.272572	−	0.962135i	$-0.587874\pi$
$587$	−13.2078	−0.545144	−0.272572	+	0.962135i	$0.587874\pi$
$588$	0	0
$589$	5.08467	0.209510
$590$	18.6948	0.769653
$591$	0	0
$592$	18.4707	0.759142
$593$	15.7161	0.645382	0.322691	−	0.946504i	$-0.395412\pi$
$593$	15.7161	0.645382	0.322691	+	0.946504i	$0.395412\pi$
$594$	0	0
$595$	−2.44277	−0.100144
$596$	−0.194376	−0.00796195
$597$	0	0
$598$	40.0970	1.63969
$599$	24.4150	0.997570	0.498785	−	0.866726i	$-0.333780\pi$
$599$	24.4150	0.997570	0.498785	+	0.866726i	$0.333780\pi$
$600$	0	0
$601$	−13.4415	−0.548291	−0.274145	−	0.961688i	$-0.588395\pi$
$601$	−13.4415	−0.548291	−0.274145	+	0.961688i	$0.588395\pi$
$602$	−34.7512	−1.41636
$603$	0	0
$604$	0.103271	0.00420204
$605$	10.7285	0.436176
$606$	0	0
$607$	11.9273	0.484114	0.242057	−	0.970262i	$-0.422178\pi$
$607$	11.9273	0.484114	0.242057	+	0.970262i	$0.422178\pi$
$608$	−0.0918970	−0.00372692
$609$	0	0
$610$	17.8571	0.723013
$611$	4.60501	0.186299
$612$	0	0
$613$	44.1079	1.78150	0.890751	−	0.454491i	$-0.150179\pi$
$613$	44.1079	1.78150	0.890751	+	0.454491i	$0.150179\pi$
$614$	12.9811	0.523876
$615$	0	0
$616$	1.77300	0.0714362
$617$	20.4987	0.825246	0.412623	−	0.910902i	$-0.364613\pi$
$617$	20.4987	0.825246	0.412623	+	0.910902i	$0.364613\pi$
$618$	0	0
$619$	−13.7570	−0.552939	−0.276470	−	0.961023i	$-0.589165\pi$
$619$	−13.7570	−0.552939	−0.276470	+	0.961023i	$0.589165\pi$
$620$	−0.0806963	−0.00324084
$621$	0	0
$622$	0.823780	0.0330306
$623$	30.2648	1.21253
$624$	0	0
$625$	11.5955	0.463822
$626$	31.3145	1.25158
$627$	0	0
$628$	0.335265	0.0133785
$629$	2.49400	0.0994422
$630$	0	0
$631$	−24.7064	−0.983545	−0.491773	−	0.870724i	$-0.663651\pi$
$631$	−24.7064	−0.983545	−0.491773	+	0.870724i	$0.663651\pi$
$632$	35.4754	1.41114
$633$	0	0
$634$	−4.23734	−0.168286
$635$	−15.3586	−0.609488
$636$	0	0
$637$	−68.1078	−2.69853
$638$	1.28687	0.0509477
$639$	0	0
$640$	−10.9159	−0.431488
$641$	32.8259	1.29654	0.648272	−	0.761409i	$-0.275492\pi$
$641$	32.8259	1.29654	0.648272	+	0.761409i	$0.275492\pi$
$642$	0	0
$643$	−29.1912	−1.15119	−0.575594	−	0.817736i	$-0.695229\pi$
$643$	−29.1912	−1.15119	−0.575594	+	0.817736i	$0.695229\pi$
$644$	−0.468815	−0.0184739
$645$	0	0
$646$	0.754475	0.0296844
$647$	−8.06642	−0.317124	−0.158562	−	0.987349i	$-0.550686\pi$
$647$	−8.06642	−0.317124	−0.158562	+	0.987349i	$0.550686\pi$
$648$	0	0
$649$	−1.81726	−0.0713336
$650$	26.2400	1.02922
$651$	0	0
$652$	0.338225	0.0132459
$653$	−31.4706	−1.23154	−0.615770	−	0.787926i	$-0.711155\pi$
$653$	−31.4706	−1.23154	−0.615770	+	0.787926i	$0.711155\pi$
$654$	0	0
$655$	−13.1310	−0.513072
$656$	−3.17240	−0.123862
$657$	0	0
$658$	6.57464	0.256306
$659$	−21.3639	−0.832220	−0.416110	−	0.909314i	$-0.636607\pi$
$659$	−21.3639	−0.832220	−0.416110	+	0.909314i	$0.636607\pi$
$660$	0	0
$661$	18.9401	0.736685	0.368343	−	0.929690i	$-0.379925\pi$
$661$	18.9401	0.736685	0.368343	+	0.929690i	$0.379925\pi$
$662$	−26.4422	−1.02771
$663$	0	0
$664$	−3.45304	−0.134004
$665$	−4.56017	−0.176836
$666$	0	0
$667$	−42.2313	−1.63520
$668$	−0.176131	−0.00681469
$669$	0	0
$670$	−18.2472	−0.704950
$671$	−1.73583	−0.0670109
$672$	0	0
$673$	10.8133	0.416821	0.208410	−	0.978041i	$-0.433171\pi$
$673$	10.8133	0.416821	0.208410	+	0.978041i	$0.433171\pi$
$674$	33.2394	1.28033
$675$	0	0
$676$	−0.133314	−0.00512747
$677$	−11.1811	−0.429725	−0.214862	−	0.976644i	$-0.568930\pi$
$677$	−11.1811	−0.429725	−0.214862	+	0.976644i	$0.568930\pi$
$678$	0	0
$679$	−41.4292	−1.58991
$680$	−1.48608	−0.0569886
$681$	0	0
$682$	−0.957856	−0.0366782
$683$	−33.2873	−1.27370	−0.636851	−	0.770987i	$-0.719763\pi$
$683$	−33.2873	−1.27370	−0.636851	+	0.770987i	$0.719763\pi$
$684$	0	0
$685$	6.09418	0.232847
$686$	−51.2159	−1.95543
$687$	0	0
$688$	−20.9695	−0.799454
$689$	48.6093	1.85187
$690$	0	0
$691$	−14.0788	−0.535584	−0.267792	−	0.963477i	$-0.586294\pi$
$691$	−14.0788	−0.535584	−0.267792	+	0.963477i	$0.586294\pi$
$692$	0.289505	0.0110053
$693$	0	0
$694$	−20.8431	−0.791193
$695$	−20.3813	−0.773106
$696$	0	0
$697$	−0.428352	−0.0162250
$698$	41.7368	1.57976
$699$	0	0
$700$	−0.306799	−0.0115959
$701$	0.623884	0.0235638	0.0117819	−	0.999931i	$-0.496250\pi$
$701$	0.623884	0.0235638	0.0117819	+	0.999931i	$0.496250\pi$
$702$	0	0
$703$	4.65581	0.175597
$704$	1.07855	0.0406494
$705$	0	0
$706$	−7.67243	−0.288756
$707$	24.8145	0.933247
$708$	0	0
$709$	−16.3186	−0.612859	−0.306430	−	0.951893i	$-0.599134\pi$
$709$	−16.3186	−0.612859	−0.306430	+	0.951893i	$0.599134\pi$
$710$	4.94751	0.185677
$711$	0	0
$712$	18.4119	0.690013
$713$	31.4340	1.17721
$714$	0	0
$715$	0.601698	0.0225022
$716$	0.193236	0.00722157
$717$	0	0
$718$	43.6193	1.62786
$719$	8.89965	0.331901	0.165951	−	0.986134i	$-0.446931\pi$
$719$	8.89965	0.331901	0.165951	+	0.986134i	$0.446931\pi$
$720$	0	0
$721$	−74.8662	−2.78816
$722$	1.40846	0.0524174
$723$	0	0
$724$	−0.192402	−0.00715056
$725$	−27.6367	−1.02640
$726$	0	0
$727$	−14.8794	−0.551845	−0.275923	−	0.961180i	$-0.588983\pi$
$727$	−14.8794	−0.551845	−0.275923	+	0.961180i	$0.588983\pi$
$728$	−61.0444	−2.26246
$729$	0	0
$730$	−11.1720	−0.413494
$731$	−2.83139	−0.104723
$732$	0	0
$733$	7.16524	0.264654	0.132327	−	0.991206i	$-0.457755\pi$
$733$	7.16524	0.264654	0.132327	+	0.991206i	$0.457755\pi$
$734$	−15.4942	−0.571903
$735$	0	0
$736$	−0.568118	−0.0209411
$737$	1.77375	0.0653368
$738$	0	0
$739$	30.7824	1.13235	0.566175	−	0.824285i	$-0.308423\pi$
$739$	30.7824	1.13235	0.566175	+	0.824285i	$0.308423\pi$
$740$	−0.0738901	−0.00271625
$741$	0	0
$742$	69.4001	2.54776
$743$	−5.15923	−0.189274	−0.0946369	−	0.995512i	$-0.530169\pi$
$743$	−5.15923	−0.189274	−0.0946369	+	0.995512i	$0.530169\pi$
$744$	0	0
$745$	−11.6885	−0.428234
$746$	−42.4801	−1.55531
$747$	0	0
$748$	0.00116394	4.25580e−5 0
$749$	11.9946	0.438275
$750$	0	0
$751$	−53.4215	−1.94938	−0.974690	−	0.223561i	$-0.928232\pi$
$751$	−53.4215	−1.94938	−0.974690	+	0.223561i	$0.928232\pi$
$752$	3.96724	0.144671
$753$	0	0
$754$	−44.3070	−1.61357
$755$	6.21004	0.226007
$756$	0	0
$757$	−21.8566	−0.794391	−0.397196	−	0.917734i	$-0.630017\pi$
$757$	−21.8566	−0.794391	−0.397196	+	0.917734i	$0.630017\pi$
$758$	9.20338	0.334282
$759$	0	0
$760$	−2.77422	−0.100632
$761$	5.27629	0.191265	0.0956327	−	0.995417i	$-0.469513\pi$
$761$	5.27629	0.191265	0.0956327	+	0.995417i	$0.469513\pi$
$762$	0	0
$763$	−26.7021	−0.966682
$764$	0.242141	0.00876036
$765$	0	0
$766$	−39.5532	−1.42912
$767$	62.5682	2.25921
$768$	0	0
$769$	15.8996	0.573356	0.286678	−	0.958027i	$-0.407449\pi$
$769$	15.8996	0.573356	0.286678	+	0.958027i	$0.407449\pi$
$770$	0.859051	0.0309581
$771$	0	0
$772$	−0.0900810	−0.00324209
$773$	−30.0601	−1.08119	−0.540593	−	0.841284i	$-0.681800\pi$
$773$	−30.0601	−1.08119	−0.540593	+	0.841284i	$0.681800\pi$
$774$	0	0
$775$	20.5708	0.738924
$776$	−25.2038	−0.904763
$777$	0	0
$778$	4.45218	0.159618
$779$	−0.799649	−0.0286504
$780$	0	0
$781$	−0.480931	−0.0172091
$782$	4.66425	0.166793
$783$	0	0
$784$	−58.6752	−2.09554
$785$	20.1607	0.719565
$786$	0	0
$787$	25.0180	0.891795	0.445898	−	0.895084i	$-0.352885\pi$
$787$	25.0180	0.891795	0.445898	+	0.895084i	$0.352885\pi$
$788$	−0.280949	−0.0100084
$789$	0	0
$790$	17.1885	0.611539
$791$	−65.8822	−2.34250
$792$	0	0
$793$	59.7646	2.12230
$794$	−27.5403	−0.977367
$795$	0	0
$796$	0.216232	0.00766416
$797$	−27.8353	−0.985976	−0.492988	−	0.870036i	$-0.664095\pi$
$797$	−27.8353	−0.985976	−0.492988	+	0.870036i	$0.664095\pi$
$798$	0	0
$799$	0.535675	0.0189508
$800$	−0.371783	−0.0131445
$801$	0	0
$802$	−8.53442	−0.301361
$803$	1.08599	0.0383238
$804$	0	0
$805$	−28.1915	−0.993620
$806$	32.9790	1.16164
$807$	0	0
$808$	15.0961	0.531080
$809$	22.2127	0.780958	0.390479	−	0.920612i	$-0.372309\pi$
$809$	22.2127	0.780958	0.390479	+	0.920612i	$0.372309\pi$
$810$	0	0
$811$	−12.4045	−0.435581	−0.217790	−	0.975996i	$-0.569885\pi$
$811$	−12.4045	−0.435581	−0.217790	+	0.975996i	$0.569885\pi$
$812$	0.518039	0.0181796
$813$	0	0
$814$	−0.877066	−0.0307412
$815$	20.3386	0.712432
$816$	0	0
$817$	−5.28565	−0.184922
$818$	−21.8985	−0.765664
$819$	0	0
$820$	0.0126908	0.000443184 0
$821$	39.4182	1.37570	0.687852	−	0.725851i	$-0.258554\pi$
$821$	39.4182	1.37570	0.687852	+	0.725851i	$0.258554\pi$
$822$	0	0
$823$	25.9212	0.903555	0.451777	−	0.892131i	$-0.350790\pi$
$823$	25.9212	0.903555	0.451777	+	0.892131i	$0.350790\pi$
$824$	−45.5455	−1.58665
$825$	0	0
$826$	89.3295	3.10817
$827$	−35.5091	−1.23477	−0.617387	−	0.786660i	$-0.711809\pi$
$827$	−35.5091	−1.23477	−0.617387	+	0.786660i	$0.711809\pi$
$828$	0	0
$829$	−10.4134	−0.361671	−0.180835	−	0.983513i	$-0.557880\pi$
$829$	−10.4134	−0.361671	−0.180835	+	0.983513i	$0.557880\pi$
$830$	−1.67306	−0.0580728
$831$	0	0
$832$	−37.1345	−1.28741
$833$	−7.92258	−0.274501
$834$	0	0
$835$	−10.5913	−0.366529
$836$	0.00217286	7.51498e−5 0
$837$	0	0
$838$	−33.0908	−1.14310
$839$	13.9023	0.479962	0.239981	−	0.970778i	$-0.422859\pi$
$839$	13.9023	0.479962	0.239981	+	0.970778i	$0.422859\pi$
$840$	0	0
$841$	17.6654	0.609152
$842$	51.0200	1.75826
$843$	0	0
$844$	0.0587754	0.00202313
$845$	−8.01665	−0.275781
$846$	0	0
$847$	51.2641	1.76146
$848$	41.8772	1.43807
$849$	0	0
$850$	3.05234	0.104695
$851$	28.7827	0.986659
$852$	0	0
$853$	8.70940	0.298204	0.149102	−	0.988822i	$-0.452362\pi$
$853$	8.70940	0.298204	0.149102	+	0.988822i	$0.452362\pi$
$854$	85.3267	2.91982
$855$	0	0
$856$	7.29704	0.249408
$857$	−28.8931	−0.986970	−0.493485	−	0.869754i	$-0.664277\pi$
$857$	−28.8931	−0.986970	−0.493485	+	0.869754i	$0.664277\pi$
$858$	0	0
$859$	37.6891	1.28594	0.642968	−	0.765893i	$-0.277703\pi$
$859$	37.6891	1.28594	0.642968	+	0.765893i	$0.277703\pi$
$860$	0.0838860	0.00286049
$861$	0	0
$862$	8.94371	0.304624
$863$	5.34349	0.181894	0.0909472	−	0.995856i	$-0.471011\pi$
$863$	5.34349	0.181894	0.0909472	+	0.995856i	$0.471011\pi$
$864$	0	0
$865$	17.4089	0.591922
$866$	19.2533	0.654252
$867$	0	0
$868$	−0.385592	−0.0130878
$869$	−1.67083	−0.0566792
$870$	0	0
$871$	−61.0701	−2.06928
$872$	−16.2445	−0.550107
$873$	0	0
$874$	8.70724	0.294527
$875$	−41.2497	−1.39450
$876$	0	0
$877$	−7.53874	−0.254565	−0.127283	−	0.991866i	$-0.540626\pi$
$877$	−7.53874	−0.254565	−0.127283	+	0.991866i	$0.540626\pi$
$878$	4.45052	0.150198
$879$	0	0
$880$	0.518366	0.0174741
$881$	48.1331	1.62165	0.810823	−	0.585291i	$-0.199020\pi$
$881$	48.1331	1.62165	0.810823	+	0.585291i	$0.199020\pi$
$882$	0	0
$883$	16.3170	0.549111	0.274555	−	0.961571i	$-0.411469\pi$
$883$	16.3170	0.549111	0.274555	+	0.961571i	$0.411469\pi$
$884$	−0.0400746	−0.00134786
$885$	0	0
$886$	15.4412	0.518757
$887$	−38.9779	−1.30875	−0.654375	−	0.756170i	$-0.727068\pi$
$887$	−38.9779	−1.30875	−0.654375	+	0.756170i	$0.727068\pi$
$888$	0	0
$889$	−73.3882	−2.46136
$890$	8.92088	0.299029
$891$	0	0
$892$	0.0438318	0.00146760
$893$	1.00000	0.0334637
$894$	0	0
$895$	11.6200	0.388412
$896$	−52.1594	−1.74252
$897$	0	0
$898$	−19.7029	−0.657495
$899$	−34.7344	−1.15846
$900$	0	0
$901$	5.65444	0.188377
$902$	0.150639	0.00501573
$903$	0	0
$904$	−40.0800	−1.33304
$905$	−11.5698	−0.384593
$906$	0	0
$907$	40.1643	1.33363	0.666817	−	0.745222i	$-0.267656\pi$
$907$	40.1643	1.33363	0.666817	+	0.745222i	$0.267656\pi$
$908$	−0.301408	−0.0100026
$909$	0	0
$910$	−29.5772	−0.980473
$911$	−35.7657	−1.18497	−0.592484	−	0.805582i	$-0.701853\pi$
$911$	−35.7657	−1.18497	−0.592484	+	0.805582i	$0.701853\pi$
$912$	0	0
$913$	0.162633	0.00538235
$914$	−26.2946	−0.869746
$915$	0	0
$916$	0.100490	0.00332028
$917$	−62.7441	−2.07199
$918$	0	0
$919$	20.7440	0.684280	0.342140	−	0.939649i	$-0.388848\pi$
$919$	20.7440	0.684280	0.342140	+	0.939649i	$0.388848\pi$
$920$	−17.1505	−0.565437
$921$	0	0
$922$	−24.0807	−0.793056
$923$	16.5585	0.545028
$924$	0	0
$925$	18.8358	0.619316
$926$	−8.92561	−0.293314
$927$	0	0
$928$	0.627768	0.0206075
$929$	−23.0224	−0.755340	−0.377670	−	0.925940i	$-0.623275\pi$
$929$	−23.0224	−0.755340	−0.377670	+	0.925940i	$0.623275\pi$
$930$	0	0
$931$	−14.7899	−0.484720
$932$	0.222125	0.00727596
$933$	0	0
$934$	−36.2527	−1.18622
$935$	0.0699920	0.00228898
$936$	0	0
$937$	−23.0155	−0.751882	−0.375941	−	0.926644i	$-0.622680\pi$
$937$	−23.0155	−0.751882	−0.375941	+	0.926644i	$0.622680\pi$
$938$	−87.1906	−2.84687
$939$	0	0
$940$	−0.0158705	−0.000517639 0
$941$	−31.9443	−1.04135	−0.520677	−	0.853754i	$-0.674320\pi$
$941$	−31.9443	−1.04135	−0.520677	+	0.853754i	$0.674320\pi$
$942$	0	0
$943$	−4.94352	−0.160983
$944$	53.9029	1.75439
$945$	0	0
$946$	0.995718	0.0323736
$947$	5.54699	0.180253	0.0901265	−	0.995930i	$-0.471273\pi$
$947$	5.54699	0.180253	0.0901265	+	0.995930i	$0.471273\pi$
$948$	0	0
$949$	−37.3907	−1.21375
$950$	5.69813	0.184872
$951$	0	0
$952$	−7.10095	−0.230143
$953$	−18.5412	−0.600610	−0.300305	−	0.953843i	$-0.597088\pi$
$953$	−18.5412	−0.600610	−0.300305	+	0.953843i	$0.597088\pi$
$954$	0	0
$955$	14.5608	0.471176
$956$	0.0192070	0.000621197 0
$957$	0	0
$958$	−44.0210	−1.42225
$959$	29.1199	0.940330
$960$	0	0
$961$	−5.14616	−0.166005
$962$	30.1974	0.973604
$963$	0	0
$964$	0.357280	0.0115072
$965$	−5.41689	−0.174376
$966$	0	0
$967$	42.2432	1.35845	0.679224	−	0.733931i	$-0.262316\pi$
$967$	42.2432	1.35845	0.679224	+	0.733931i	$0.262316\pi$
$968$	31.1870	1.00239
$969$	0	0
$970$	−12.2117	−0.392094
$971$	56.0389	1.79837	0.899187	−	0.437565i	$-0.144159\pi$
$971$	56.0389	1.79837	0.899187	+	0.437565i	$0.144159\pi$
$972$	0	0
$973$	−97.3880	−3.12212
$974$	28.3568	0.908612
$975$	0	0
$976$	51.4876	1.64808
$977$	60.4861	1.93512	0.967561	−	0.252638i	$-0.0812981\pi$
$977$	60.4861	1.93512	0.967561	+	0.252638i	$0.0812981\pi$
$978$	0	0
$979$	−0.867169	−0.0277148
$980$	0.234724	0.00749797
$981$	0	0
$982$	−25.2567	−0.805974
$983$	27.9095	0.890175	0.445088	−	0.895487i	$-0.353173\pi$
$983$	27.9095	0.890175	0.445088	+	0.895487i	$0.353173\pi$
$984$	0	0
$985$	−16.8945	−0.538302
$986$	−5.15398	−0.164136
$987$	0	0
$988$	−0.0748115	−0.00238007
$989$	−32.6765	−1.03905
$990$	0	0
$991$	1.59793	0.0507599	0.0253800	−	0.999678i	$-0.491920\pi$
$991$	1.59793	0.0507599	0.0253800	+	0.999678i	$0.491920\pi$
$992$	−0.467266	−0.0148357
$993$	0	0
$994$	23.6407	0.749838
$995$	13.0028	0.412217
$996$	0	0
$997$	43.8255	1.38797	0.693983	−	0.719991i	$-0.255854\pi$
$997$	43.8255	1.38797	0.693983	+	0.719991i	$0.255854\pi$
$998$	6.41690	0.203123
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.o.1.13		18
3.2	odd	2		893.2.a.c.1.6	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.c.1.6	✓	18	3.2	odd	2
8037.2.a.o.1.13		18	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+1.40846 q^{2} -0.0162457 q^{4} -0.976908 q^{5} -4.66797 q^{7} -2.83980 q^{8} +O(q^{10})\)	\(q+1.40846 q^{2} -0.0162457 q^{4} -0.976908 q^{5} -4.66797 q^{7} -2.83980 q^{8} -1.37593 q^{10} +0.133750 q^{11} -4.60501 q^{13} -6.57464 q^{14} -3.96724 q^{16} -0.535675 q^{17} -1.00000 q^{19} +0.0158705 q^{20} +0.188381 q^{22} -6.18211 q^{23} -4.04565 q^{25} -6.48597 q^{26} +0.0758342 q^{28} +6.83121 q^{29} -5.08467 q^{31} +0.0918970 q^{32} -0.754475 q^{34} +4.56017 q^{35} -4.65581 q^{37} -1.40846 q^{38} +2.77422 q^{40} +0.799649 q^{41} +5.28565 q^{43} -0.00217286 q^{44} -8.70724 q^{46} -1.00000 q^{47} +14.7899 q^{49} -5.69813 q^{50} +0.0748115 q^{52} -10.5557 q^{53} -0.130661 q^{55} +13.2561 q^{56} +9.62147 q^{58} -13.5870 q^{59} -12.9782 q^{61} -7.16154 q^{62} +8.06392 q^{64} +4.49867 q^{65} +13.2617 q^{67} +0.00870239 q^{68} +6.42282 q^{70} -3.59575 q^{71} +8.11956 q^{73} -6.55751 q^{74} +0.0162457 q^{76} -0.624340 q^{77} -12.4922 q^{79} +3.87563 q^{80} +1.12627 q^{82} +1.21594 q^{83} +0.523305 q^{85} +7.44462 q^{86} -0.379823 q^{88} -6.48351 q^{89} +21.4960 q^{91} +0.100432 q^{92} -1.40846 q^{94} +0.976908 q^{95} +8.87520 q^{97} +20.8310 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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