LMFDB - Embedded newform 8037.2.a.w.1.2

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:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.45233 q^{2} +4.01392 q^{4} -0.994616 q^{5} -2.94476 q^{7} -4.93878 q^{8} +O(q^{10})$	\(q-2.45233 q^{2} +4.01392 q^{4} -0.994616 q^{5} -2.94476 q^{7} -4.93878 q^{8} +2.43912 q^{10} -2.20273 q^{11} -6.02261 q^{13} +7.22151 q^{14} +4.08369 q^{16} +5.53068 q^{17} -1.00000 q^{19} -3.99230 q^{20} +5.40181 q^{22} -2.53451 q^{23} -4.01074 q^{25} +14.7694 q^{26} -11.8200 q^{28} -3.00985 q^{29} -4.37755 q^{31} -0.136977 q^{32} -13.5630 q^{34} +2.92890 q^{35} +3.76888 q^{37} +2.45233 q^{38} +4.91219 q^{40} -3.76866 q^{41} -3.40528 q^{43} -8.84157 q^{44} +6.21546 q^{46} +1.00000 q^{47} +1.67158 q^{49} +9.83565 q^{50} -24.1743 q^{52} -0.817034 q^{53} +2.19087 q^{55} +14.5435 q^{56} +7.38113 q^{58} +1.26956 q^{59} -5.16632 q^{61} +10.7352 q^{62} -7.83146 q^{64} +5.99019 q^{65} -1.75419 q^{67} +22.1997 q^{68} -7.18263 q^{70} +2.59327 q^{71} -13.5694 q^{73} -9.24253 q^{74} -4.01392 q^{76} +6.48650 q^{77} -7.14735 q^{79} -4.06170 q^{80} +9.24200 q^{82} -8.94391 q^{83} -5.50090 q^{85} +8.35087 q^{86} +10.8788 q^{88} -9.24878 q^{89} +17.7351 q^{91} -10.1733 q^{92} -2.45233 q^{94} +0.994616 q^{95} -2.97847 q^{97} -4.09927 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * q^97 + 45 * 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 $

$2$	−2.45233	−1.73406	−0.867029	−	0.498258i	$-0.833973\pi$
$2$	−2.45233	−1.73406	−0.867029	+	0.498258i	$0.833973\pi$
$3$	0	0
$3$	0	0
$4$	4.01392	2.00696
$5$	−0.994616	−0.444806	−0.222403	−	0.974955i	$-0.571390\pi$
$5$	−0.994616	−0.444806	−0.222403	+	0.974955i	$0.571390\pi$
$6$	0	0
$7$	−2.94476	−1.11301	−0.556506	−	0.830843i	$-0.687859\pi$
$7$	−2.94476	−1.11301	−0.556506	+	0.830843i	$0.687859\pi$
$8$	−4.93878	−1.74612
$9$	0	0
$10$	2.43912	0.771319
$11$	−2.20273	−0.664148	−0.332074	−	0.943253i	$-0.607748\pi$
$11$	−2.20273	−0.664148	−0.332074	+	0.943253i	$0.607748\pi$
$12$	0	0
$13$	−6.02261	−1.67037	−0.835186	−	0.549967i	$-0.814640\pi$
$13$	−6.02261	−1.67037	−0.835186	+	0.549967i	$0.814640\pi$
$14$	7.22151	1.93003
$15$	0	0
$16$	4.08369	1.02092
$17$	5.53068	1.34139	0.670694	−	0.741734i	$-0.265997\pi$
$17$	5.53068	1.34139	0.670694	+	0.741734i	$0.265997\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−3.99230	−0.892706
$21$	0	0
$22$	5.40181	1.15167
$23$	−2.53451	−0.528483	−0.264241	−	0.964457i	$-0.585122\pi$
$23$	−2.53451	−0.528483	−0.264241	+	0.964457i	$0.585122\pi$
$24$	0	0
$25$	−4.01074	−0.802148
$26$	14.7694	2.89652
$27$	0	0
$28$	−11.8200	−2.23377
$29$	−3.00985	−0.558914	−0.279457	−	0.960158i	$-0.590155\pi$
$29$	−3.00985	−0.558914	−0.279457	+	0.960158i	$0.590155\pi$
$30$	0	0
$31$	−4.37755	−0.786230	−0.393115	−	0.919489i	$-0.628603\pi$
$31$	−4.37755	−0.786230	−0.393115	+	0.919489i	$0.628603\pi$
$32$	−0.136977	−0.0242144
$33$	0	0
$34$	−13.5630	−2.32604
$35$	2.92890	0.495075
$36$	0	0
$37$	3.76888	0.619600	0.309800	−	0.950802i	$-0.399738\pi$
$37$	3.76888	0.619600	0.309800	+	0.950802i	$0.399738\pi$
$38$	2.45233	0.397820
$39$	0	0
$40$	4.91219	0.776686
$41$	−3.76866	−0.588566	−0.294283	−	0.955718i	$-0.595081\pi$
$41$	−3.76866	−0.588566	−0.294283	+	0.955718i	$0.595081\pi$
$42$	0	0
$43$	−3.40528	−0.519301	−0.259650	−	0.965703i	$-0.583607\pi$
$43$	−3.40528	−0.519301	−0.259650	+	0.965703i	$0.583607\pi$
$44$	−8.84157	−1.33292
$45$	0	0
$46$	6.21546	0.916420
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	1.67158	0.238798
$50$	9.83565	1.39097
$51$	0	0
$52$	−24.1743	−3.35237
$53$	−0.817034	−0.112228	−0.0561141	−	0.998424i	$-0.517871\pi$
$53$	−0.817034	−0.112228	−0.0561141	+	0.998424i	$0.517871\pi$
$54$	0	0
$55$	2.19087	0.295417
$56$	14.5435	1.94346
$57$	0	0
$58$	7.38113	0.969190
$59$	1.26956	0.165282	0.0826412	−	0.996579i	$-0.473664\pi$
$59$	1.26956	0.165282	0.0826412	+	0.996579i	$0.473664\pi$
$60$	0	0
$61$	−5.16632	−0.661480	−0.330740	−	0.943722i	$-0.607298\pi$
$61$	−5.16632	−0.661480	−0.330740	+	0.943722i	$0.607298\pi$
$62$	10.7352	1.36337
$63$	0	0
$64$	−7.83146	−0.978933
$65$	5.99019	0.742991
$66$	0	0
$67$	−1.75419	−0.214309	−0.107154	−	0.994242i	$-0.534174\pi$
$67$	−1.75419	−0.214309	−0.107154	+	0.994242i	$0.534174\pi$
$68$	22.1997	2.69211
$69$	0	0
$70$	−7.18263	−0.858488
$71$	2.59327	0.307764	0.153882	−	0.988089i	$-0.450822\pi$
$71$	2.59327	0.307764	0.153882	+	0.988089i	$0.450822\pi$
$72$	0	0
$73$	−13.5694	−1.58818	−0.794090	−	0.607800i	$-0.792052\pi$
$73$	−13.5694	−1.58818	−0.794090	+	0.607800i	$0.792052\pi$
$74$	−9.24253	−1.07442
$75$	0	0
$76$	−4.01392	−0.460428
$77$	6.48650	0.739205
$78$	0	0
$79$	−7.14735	−0.804140	−0.402070	−	0.915609i	$-0.631709\pi$
$79$	−7.14735	−0.804140	−0.402070	+	0.915609i	$0.631709\pi$
$80$	−4.06170	−0.454112
$81$	0	0
$82$	9.24200	1.02061
$83$	−8.94391	−0.981722	−0.490861	−	0.871238i	$-0.663318\pi$
$83$	−8.94391	−0.981722	−0.490861	+	0.871238i	$0.663318\pi$
$84$	0	0
$85$	−5.50090	−0.596657
$86$	8.35087	0.900497
$87$	0	0
$88$	10.8788	1.15968
$89$	−9.24878	−0.980369	−0.490184	−	0.871619i	$-0.663071\pi$
$89$	−9.24878	−0.980369	−0.490184	+	0.871619i	$0.663071\pi$
$90$	0	0
$91$	17.7351	1.85915
$92$	−10.1733	−1.06064
$93$	0	0
$94$	−2.45233	−0.252938
$95$	0.994616	0.102045
$96$	0	0
$97$	−2.97847	−0.302417	−0.151209	−	0.988502i	$-0.548317\pi$
$97$	−2.97847	−0.302417	−0.151209	+	0.988502i	$0.548317\pi$
$98$	−4.09927	−0.414089
$99$	0	0
$100$	−16.0988	−1.60988
$101$	−3.08960	−0.307427	−0.153714	−	0.988115i	$-0.549123\pi$
$101$	−3.08960	−0.307427	−0.153714	+	0.988115i	$0.549123\pi$
$102$	0	0
$103$	−13.4983	−1.33003	−0.665016	−	0.746830i	$-0.731575\pi$
$103$	−13.4983	−1.33003	−0.665016	+	0.746830i	$0.731575\pi$
$104$	29.7444	2.91668
$105$	0	0
$106$	2.00364	0.194610
$107$	−1.68582	−0.162974	−0.0814872	−	0.996674i	$-0.525967\pi$
$107$	−1.68582	−0.162974	−0.0814872	+	0.996674i	$0.525967\pi$
$108$	0	0
$109$	−18.0224	−1.72623	−0.863115	−	0.505007i	$-0.831490\pi$
$109$	−18.0224	−1.72623	−0.863115	+	0.505007i	$0.831490\pi$
$110$	−5.37273	−0.512270
$111$	0	0
$112$	−12.0255	−1.13630
$113$	14.2566	1.34114	0.670572	−	0.741844i	$-0.266049\pi$
$113$	14.2566	1.34114	0.670572	+	0.741844i	$0.266049\pi$
$114$	0	0
$115$	2.52087	0.235072
$116$	−12.0813	−1.12172
$117$	0	0
$118$	−3.11337	−0.286609
$119$	−16.2865	−1.49298
$120$	0	0
$121$	−6.14799	−0.558908
$122$	12.6695	1.14704
$123$	0	0
$124$	−17.5711	−1.57793
$125$	8.96222	0.801606
$126$	0	0
$127$	−7.83612	−0.695343	−0.347671	−	0.937616i	$-0.613028\pi$
$127$	−7.83612	−0.695343	−0.347671	+	0.937616i	$0.613028\pi$
$128$	19.4793	1.72174
$129$	0	0
$130$	−14.6899	−1.28839
$131$	−12.7296	−1.11219	−0.556095	−	0.831119i	$-0.687701\pi$
$131$	−12.7296	−1.11219	−0.556095	+	0.831119i	$0.687701\pi$
$132$	0	0
$133$	2.94476	0.255343
$134$	4.30186	0.371624
$135$	0	0
$136$	−27.3148	−2.34223
$137$	−1.57803	−0.134820	−0.0674099	−	0.997725i	$-0.521474\pi$
$137$	−1.57803	−0.134820	−0.0674099	+	0.997725i	$0.521474\pi$
$138$	0	0
$139$	13.6399	1.15692	0.578461	−	0.815710i	$-0.303653\pi$
$139$	13.6399	1.15692	0.578461	+	0.815710i	$0.303653\pi$
$140$	11.7564	0.993594
$141$	0	0
$142$	−6.35954	−0.533681
$143$	13.2662	1.10937
$144$	0	0
$145$	2.99364	0.248608
$146$	33.2767	2.75400
$147$	0	0
$148$	15.1280	1.24351
$149$	−23.5732	−1.93119	−0.965597	−	0.260045i	$-0.916263\pi$
$149$	−23.5732	−1.93119	−0.965597	+	0.260045i	$0.916263\pi$
$150$	0	0
$151$	−1.36535	−0.111111	−0.0555554	−	0.998456i	$-0.517693\pi$
$151$	−1.36535	−0.111111	−0.0555554	+	0.998456i	$0.517693\pi$
$152$	4.93878	0.400588
$153$	0	0
$154$	−15.9070	−1.28182
$155$	4.35398	0.349720
$156$	0	0
$157$	−5.79149	−0.462211	−0.231105	−	0.972929i	$-0.574234\pi$
$157$	−5.79149	−0.462211	−0.231105	+	0.972929i	$0.574234\pi$
$158$	17.5277	1.39443
$159$	0	0
$160$	0.136240	0.0107707
$161$	7.46352	0.588208
$162$	0	0
$163$	12.0440	0.943359	0.471680	−	0.881770i	$-0.343648\pi$
$163$	12.0440	0.943359	0.471680	+	0.881770i	$0.343648\pi$
$164$	−15.1271	−1.18123
$165$	0	0
$166$	21.9334	1.70236
$167$	−7.06449	−0.546666	−0.273333	−	0.961919i	$-0.588126\pi$
$167$	−7.06449	−0.546666	−0.273333	+	0.961919i	$0.588126\pi$
$168$	0	0
$169$	23.2719	1.79014
$170$	13.4900	1.03464
$171$	0	0
$172$	−13.6685	−1.04221
$173$	−1.31177	−0.0997322	−0.0498661	−	0.998756i	$-0.515879\pi$
$173$	−1.31177	−0.0997322	−0.0498661	+	0.998756i	$0.515879\pi$
$174$	0	0
$175$	11.8106	0.892801
$176$	−8.99525	−0.678043
$177$	0	0
$178$	22.6810	1.70002
$179$	−2.64889	−0.197988	−0.0989938	−	0.995088i	$-0.531562\pi$
$179$	−2.64889	−0.197988	−0.0989938	+	0.995088i	$0.531562\pi$
$180$	0	0
$181$	20.0783	1.49241	0.746205	−	0.665716i	$-0.231874\pi$
$181$	20.0783	1.49241	0.746205	+	0.665716i	$0.231874\pi$
$182$	−43.4923	−3.22387
$183$	0	0
$184$	12.5174	0.922796
$185$	−3.74859	−0.275602
$186$	0	0
$187$	−12.1826	−0.890879
$188$	4.01392	0.292745
$189$	0	0
$190$	−2.43912	−0.176953
$191$	−22.3493	−1.61714	−0.808571	−	0.588399i	$-0.799759\pi$
$191$	−22.3493	−1.61714	−0.808571	+	0.588399i	$0.799759\pi$
$192$	0	0
$193$	8.26856	0.595184	0.297592	−	0.954693i	$-0.403817\pi$
$193$	8.26856	0.595184	0.297592	+	0.954693i	$0.403817\pi$
$194$	7.30418	0.524409
$195$	0	0
$196$	6.70960	0.479257
$197$	−6.15536	−0.438551	−0.219275	−	0.975663i	$-0.570369\pi$
$197$	−6.15536	−0.438551	−0.219275	+	0.975663i	$0.570369\pi$
$198$	0	0
$199$	−14.3198	−1.01510	−0.507551	−	0.861622i	$-0.669449\pi$
$199$	−14.3198	−1.01510	−0.507551	+	0.861622i	$0.669449\pi$
$200$	19.8082	1.40065
$201$	0	0
$202$	7.57672	0.533096
$203$	8.86326	0.622079
$204$	0	0
$205$	3.74837	0.261798
$206$	33.1024	2.30635
$207$	0	0
$208$	−24.5945	−1.70532
$209$	2.20273	0.152366
$210$	0	0
$211$	15.5014	1.06716	0.533581	−	0.845749i	$-0.320846\pi$
$211$	15.5014	1.06716	0.533581	+	0.845749i	$0.320846\pi$
$212$	−3.27951	−0.225237
$213$	0	0
$214$	4.13419	0.282607
$215$	3.38695	0.230988
$216$	0	0
$217$	12.8908	0.875085
$218$	44.1968	2.99338
$219$	0	0
$220$	8.79396	0.592889
$221$	−33.3092	−2.24062
$222$	0	0
$223$	−4.33365	−0.290203	−0.145101	−	0.989417i	$-0.546351\pi$
$223$	−4.33365	−0.290203	−0.145101	+	0.989417i	$0.546351\pi$
$224$	0.403364	0.0269509
$225$	0	0
$226$	−34.9618	−2.32562
$227$	22.9756	1.52494	0.762471	−	0.647022i	$-0.223986\pi$
$227$	22.9756	1.52494	0.762471	+	0.647022i	$0.223986\pi$
$228$	0	0
$229$	−17.6722	−1.16781	−0.583906	−	0.811821i	$-0.698476\pi$
$229$	−17.6722	−1.16781	−0.583906	+	0.811821i	$0.698476\pi$
$230$	−6.18200	−0.407629
$231$	0	0
$232$	14.8650	0.975934
$233$	16.2649	1.06555	0.532775	−	0.846257i	$-0.321149\pi$
$233$	16.2649	1.06555	0.532775	+	0.846257i	$0.321149\pi$
$234$	0	0
$235$	−0.994616	−0.0648816
$236$	5.09590	0.331715
$237$	0	0
$238$	39.9399	2.58892
$239$	2.47454	0.160065	0.0800323	−	0.996792i	$-0.474498\pi$
$239$	2.47454	0.160065	0.0800323	+	0.996792i	$0.474498\pi$
$240$	0	0
$241$	−2.87144	−0.184966	−0.0924829	−	0.995714i	$-0.529480\pi$
$241$	−2.87144	−0.184966	−0.0924829	+	0.995714i	$0.529480\pi$
$242$	15.0769	0.969179
$243$	0	0
$244$	−20.7372	−1.32756
$245$	−1.66258	−0.106219
$246$	0	0
$247$	6.02261	0.383210
$248$	21.6197	1.37286
$249$	0	0
$250$	−21.9783	−1.39003
$251$	9.07634	0.572893	0.286447	−	0.958096i	$-0.407526\pi$
$251$	9.07634	0.572893	0.286447	+	0.958096i	$0.407526\pi$
$252$	0	0
$253$	5.58285	0.350991
$254$	19.2167	1.20577
$255$	0	0
$256$	−32.1067	−2.00667
$257$	10.7748	0.672116	0.336058	−	0.941841i	$-0.390906\pi$
$257$	10.7748	0.672116	0.336058	+	0.941841i	$0.390906\pi$
$258$	0	0
$259$	−11.0984	−0.689622
$260$	24.0441	1.49115
$261$	0	0
$262$	31.2172	1.92860
$263$	−2.17989	−0.134418	−0.0672090	−	0.997739i	$-0.521409\pi$
$263$	−2.17989	−0.134418	−0.0672090	+	0.997739i	$0.521409\pi$
$264$	0	0
$265$	0.812635	0.0499198
$266$	−7.22151	−0.442779
$267$	0	0
$268$	−7.04119	−0.430109
$269$	−0.589002	−0.0359121	−0.0179560	−	0.999839i	$-0.505716\pi$
$269$	−0.589002	−0.0359121	−0.0179560	+	0.999839i	$0.505716\pi$
$270$	0	0
$271$	29.5777	1.79672	0.898358	−	0.439263i	$-0.144760\pi$
$271$	29.5777	1.79672	0.898358	+	0.439263i	$0.144760\pi$
$272$	22.5856	1.36945
$273$	0	0
$274$	3.86984	0.233786
$275$	8.83457	0.532745
$276$	0	0
$277$	18.7338	1.12560	0.562802	−	0.826592i	$-0.309723\pi$
$277$	18.7338	1.12560	0.562802	+	0.826592i	$0.309723\pi$
$278$	−33.4496	−2.00617
$279$	0	0
$280$	−14.4652	−0.864461
$281$	−19.6177	−1.17029	−0.585146	−	0.810928i	$-0.698963\pi$
$281$	−19.6177	−1.17029	−0.585146	+	0.810928i	$0.698963\pi$
$282$	0	0
$283$	2.58878	0.153887	0.0769436	−	0.997035i	$-0.475484\pi$
$283$	2.58878	0.153887	0.0769436	+	0.997035i	$0.475484\pi$
$284$	10.4091	0.617669
$285$	0	0
$286$	−32.5330	−1.92372
$287$	11.0978	0.655082
$288$	0	0
$289$	13.5884	0.799320
$290$	−7.34139	−0.431101
$291$	0	0
$292$	−54.4665	−3.18741
$293$	−25.3347	−1.48007	−0.740035	−	0.672569i	$-0.765191\pi$
$293$	−25.3347	−1.48007	−0.740035	+	0.672569i	$0.765191\pi$
$294$	0	0
$295$	−1.26272	−0.0735186
$296$	−18.6137	−1.08190
$297$	0	0
$298$	57.8093	3.34880
$299$	15.2644	0.882763
$300$	0	0
$301$	10.0277	0.577988
$302$	3.34829	0.192673
$303$	0	0
$304$	−4.08369	−0.234216
$305$	5.13851	0.294230
$306$	0	0
$307$	−26.7338	−1.52578	−0.762889	−	0.646529i	$-0.776220\pi$
$307$	−26.7338	−1.52578	−0.762889	+	0.646529i	$0.776220\pi$
$308$	26.0363	1.48355
$309$	0	0
$310$	−10.6774	−0.606435
$311$	29.8568	1.69302	0.846511	−	0.532372i	$-0.178699\pi$
$311$	29.8568	1.69302	0.846511	+	0.532372i	$0.178699\pi$
$312$	0	0
$313$	−26.1921	−1.48046	−0.740231	−	0.672352i	$-0.765284\pi$
$313$	−26.1921	−1.48046	−0.740231	+	0.672352i	$0.765284\pi$
$314$	14.2026	0.801501
$315$	0	0
$316$	−28.6889	−1.61388
$317$	7.17886	0.403205	0.201603	−	0.979467i	$-0.435385\pi$
$317$	7.17886	0.403205	0.201603	+	0.979467i	$0.435385\pi$
$318$	0	0
$319$	6.62987	0.371202
$320$	7.78930	0.435435
$321$	0	0
$322$	−18.3030	−1.01999
$323$	−5.53068	−0.307735
$324$	0	0
$325$	24.1551	1.33989
$326$	−29.5359	−1.63584
$327$	0	0
$328$	18.6126	1.02771
$329$	−2.94476	−0.162350
$330$	0	0
$331$	−2.93719	−0.161442	−0.0807212	−	0.996737i	$-0.525722\pi$
$331$	−2.93719	−0.161442	−0.0807212	+	0.996737i	$0.525722\pi$
$332$	−35.9001	−1.97027
$333$	0	0
$334$	17.3244	0.947951
$335$	1.74475	0.0953258
$336$	0	0
$337$	−10.7279	−0.584384	−0.292192	−	0.956360i	$-0.594385\pi$
$337$	−10.7279	−0.584384	−0.292192	+	0.956360i	$0.594385\pi$
$338$	−57.0703	−3.10421
$339$	0	0
$340$	−22.0802	−1.19746
$341$	9.64255	0.522173
$342$	0	0
$343$	15.6909	0.847228
$344$	16.8179	0.906763
$345$	0	0
$346$	3.21690	0.172941
$347$	−4.80751	−0.258081	−0.129040	−	0.991639i	$-0.541190\pi$
$347$	−4.80751	−0.258081	−0.129040	+	0.991639i	$0.541190\pi$
$348$	0	0
$349$	−31.9178	−1.70852	−0.854260	−	0.519847i	$-0.825989\pi$
$349$	−31.9178	−1.70852	−0.854260	+	0.519847i	$0.825989\pi$
$350$	−28.9636	−1.54817
$351$	0	0
$352$	0.301723	0.0160819
$353$	20.0923	1.06941	0.534703	−	0.845040i	$-0.320424\pi$
$353$	20.0923	1.06941	0.534703	+	0.845040i	$0.320424\pi$
$354$	0	0
$355$	−2.57930	−0.136895
$356$	−37.1238	−1.96756
$357$	0	0
$358$	6.49596	0.343322
$359$	−30.1324	−1.59032	−0.795162	−	0.606397i	$-0.792614\pi$
$359$	−30.1324	−1.59032	−0.795162	+	0.606397i	$0.792614\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−49.2386	−2.58792
$363$	0	0
$364$	71.1873	3.73123
$365$	13.4964	0.706432
$366$	0	0
$367$	−4.27422	−0.223113	−0.111556	−	0.993758i	$-0.535584\pi$
$367$	−4.27422	−0.223113	−0.111556	+	0.993758i	$0.535584\pi$
$368$	−10.3502	−0.539540
$369$	0	0
$370$	9.19276	0.477909
$371$	2.40597	0.124912
$372$	0	0
$373$	26.7585	1.38550	0.692752	−	0.721176i	$-0.256398\pi$
$373$	26.7585	1.38550	0.692752	+	0.721176i	$0.256398\pi$
$374$	29.8757	1.54484
$375$	0	0
$376$	−4.93878	−0.254698
$377$	18.1271	0.933595
$378$	0	0
$379$	11.9725	0.614986	0.307493	−	0.951550i	$-0.400510\pi$
$379$	11.9725	0.614986	0.307493	+	0.951550i	$0.400510\pi$
$380$	3.99230	0.204801
$381$	0	0
$382$	54.8079	2.80422
$383$	5.71668	0.292109	0.146054	−	0.989277i	$-0.453343\pi$
$383$	5.71668	0.292109	0.146054	+	0.989277i	$0.453343\pi$
$384$	0	0
$385$	−6.45157	−0.328803
$386$	−20.2772	−1.03208
$387$	0	0
$388$	−11.9553	−0.606939
$389$	−34.0008	−1.72391	−0.861954	−	0.506986i	$-0.830759\pi$
$389$	−34.0008	−1.72391	−0.861954	+	0.506986i	$0.830759\pi$
$390$	0	0
$391$	−14.0176	−0.708900
$392$	−8.25559	−0.416970
$393$	0	0
$394$	15.0950	0.760473
$395$	7.10887	0.357686
$396$	0	0
$397$	14.9170	0.748662	0.374331	−	0.927295i	$-0.377872\pi$
$397$	14.9170	0.748662	0.374331	+	0.927295i	$0.377872\pi$
$398$	35.1168	1.76025
$399$	0	0
$400$	−16.3786	−0.818930
$401$	−6.68969	−0.334067	−0.167033	−	0.985951i	$-0.553419\pi$
$401$	−6.68969	−0.334067	−0.167033	+	0.985951i	$0.553419\pi$
$402$	0	0
$403$	26.3643	1.31330
$404$	−12.4014	−0.616993
$405$	0	0
$406$	−21.7356	−1.07872
$407$	−8.30182	−0.411506
$408$	0	0
$409$	−9.60770	−0.475070	−0.237535	−	0.971379i	$-0.576340\pi$
$409$	−9.60770	−0.475070	−0.237535	+	0.971379i	$0.576340\pi$
$410$	−9.19224	−0.453972
$411$	0	0
$412$	−54.1812	−2.66932
$413$	−3.73854	−0.183961
$414$	0	0
$415$	8.89576	0.436675
$416$	0.824960	0.0404470
$417$	0	0
$418$	−5.40181	−0.264211
$419$	4.40436	0.215167	0.107583	−	0.994196i	$-0.465689\pi$
$419$	4.40436	0.215167	0.107583	+	0.994196i	$0.465689\pi$
$420$	0	0
$421$	4.17670	0.203560	0.101780	−	0.994807i	$-0.467546\pi$
$421$	4.17670	0.203560	0.101780	+	0.994807i	$0.467546\pi$
$422$	−38.0146	−1.85052
$423$	0	0
$424$	4.03515	0.195964
$425$	−22.1821	−1.07599
$426$	0	0
$427$	15.2136	0.736236
$428$	−6.76674	−0.327083
$429$	0	0
$430$	−8.30591	−0.400546
$431$	−3.40902	−0.164207	−0.0821035	−	0.996624i	$-0.526164\pi$
$431$	−3.40902	−0.164207	−0.0821035	+	0.996624i	$0.526164\pi$
$432$	0	0
$433$	10.3971	0.499654	0.249827	−	0.968291i	$-0.419626\pi$
$433$	10.3971	0.499654	0.249827	+	0.968291i	$0.419626\pi$
$434$	−31.6125	−1.51745
$435$	0	0
$436$	−72.3403	−3.46447
$437$	2.53451	0.121242
$438$	0	0
$439$	8.35438	0.398733	0.199366	−	0.979925i	$-0.436112\pi$
$439$	8.35438	0.398733	0.199366	+	0.979925i	$0.436112\pi$
$440$	−10.8202	−0.515834
$441$	0	0
$442$	81.6850	3.88536
$443$	11.0525	0.525121	0.262560	−	0.964916i	$-0.415433\pi$
$443$	11.0525	0.525121	0.262560	+	0.964916i	$0.415433\pi$
$444$	0	0
$445$	9.19898	0.436074
$446$	10.6275	0.503229
$447$	0	0
$448$	23.0617	1.08956
$449$	−3.84227	−0.181328	−0.0906640	−	0.995882i	$-0.528899\pi$
$449$	−3.84227	−0.181328	−0.0906640	+	0.995882i	$0.528899\pi$
$450$	0	0
$451$	8.30134	0.390895
$452$	57.2246	2.69162
$453$	0	0
$454$	−56.3437	−2.64434
$455$	−17.6396	−0.826959
$456$	0	0
$457$	−32.6929	−1.52931	−0.764656	−	0.644439i	$-0.777091\pi$
$457$	−32.6929	−1.52931	−0.764656	+	0.644439i	$0.777091\pi$
$458$	43.3380	2.02505
$459$	0	0
$460$	10.1186	0.471780
$461$	3.28437	0.152968	0.0764841	−	0.997071i	$-0.475631\pi$
$461$	3.28437	0.152968	0.0764841	+	0.997071i	$0.475631\pi$
$462$	0	0
$463$	−14.2545	−0.662462	−0.331231	−	0.943550i	$-0.607464\pi$
$463$	−14.2545	−0.662462	−0.331231	+	0.943550i	$0.607464\pi$
$464$	−12.2913	−0.570608
$465$	0	0
$466$	−39.8869	−1.84773
$467$	−14.2926	−0.661383	−0.330691	−	0.943739i	$-0.607282\pi$
$467$	−14.2926	−0.661383	−0.330691	+	0.943739i	$0.607282\pi$
$468$	0	0
$469$	5.16567	0.238529
$470$	2.43912	0.112508
$471$	0	0
$472$	−6.27007	−0.288604
$473$	7.50091	0.344892
$474$	0	0
$475$	4.01074	0.184025
$476$	−65.3727	−2.99635
$477$	0	0
$478$	−6.06838	−0.277561
$479$	38.5327	1.76061	0.880303	−	0.474412i	$-0.157339\pi$
$479$	38.5327	1.76061	0.880303	+	0.474412i	$0.157339\pi$
$480$	0	0
$481$	−22.6985	−1.03496
$482$	7.04172	0.320741
$483$	0	0
$484$	−24.6775	−1.12170
$485$	2.96243	0.134517
$486$	0	0
$487$	−14.6602	−0.664316	−0.332158	−	0.943224i	$-0.607777\pi$
$487$	−14.6602	−0.664316	−0.332158	+	0.943224i	$0.607777\pi$
$488$	25.5154	1.15503
$489$	0	0
$490$	4.07720	0.184189
$491$	5.92445	0.267367	0.133683	−	0.991024i	$-0.457320\pi$
$491$	5.92445	0.267367	0.133683	+	0.991024i	$0.457320\pi$
$492$	0	0
$493$	−16.6465	−0.749721
$494$	−14.7694	−0.664508
$495$	0	0
$496$	−17.8765	−0.802680
$497$	−7.63653	−0.342545
$498$	0	0
$499$	11.8173	0.529017	0.264508	−	0.964383i	$-0.414790\pi$
$499$	11.8173	0.529017	0.264508	+	0.964383i	$0.414790\pi$
$500$	35.9736	1.60879
$501$	0	0
$502$	−22.2582	−0.993430
$503$	−8.40475	−0.374749	−0.187375	−	0.982289i	$-0.559998\pi$
$503$	−8.40475	−0.374749	−0.187375	+	0.982289i	$0.559998\pi$
$504$	0	0
$505$	3.07297	0.136745
$506$	−13.6910	−0.608638
$507$	0	0
$508$	−31.4535	−1.39552
$509$	−32.0158	−1.41907	−0.709537	−	0.704668i	$-0.751096\pi$
$509$	−32.0158	−1.41907	−0.709537	+	0.704668i	$0.751096\pi$
$510$	0	0
$511$	39.9586	1.76767
$512$	39.7775	1.75793
$513$	0	0
$514$	−26.4235	−1.16549
$515$	13.4257	0.591605
$516$	0	0
$517$	−2.20273	−0.0968759
$518$	27.2170	1.19585
$519$	0	0
$520$	−29.5842	−1.29735
$521$	28.4980	1.24852	0.624260	−	0.781217i	$-0.285401\pi$
$521$	28.4980	1.24852	0.624260	+	0.781217i	$0.285401\pi$
$522$	0	0
$523$	6.96848	0.304710	0.152355	−	0.988326i	$-0.451314\pi$
$523$	6.96848	0.304710	0.152355	+	0.988326i	$0.451314\pi$
$524$	−51.0955	−2.23212
$525$	0	0
$526$	5.34582	0.233089
$527$	−24.2108	−1.05464
$528$	0	0
$529$	−16.5762	−0.720706
$530$	−1.99285	−0.0865638
$531$	0	0
$532$	11.8200	0.512462
$533$	22.6972	0.983125
$534$	0	0
$535$	1.67674	0.0724919
$536$	8.66358	0.374210
$537$	0	0
$538$	1.44443	0.0622736
$539$	−3.68205	−0.158597
$540$	0	0
$541$	−30.9331	−1.32992	−0.664960	−	0.746879i	$-0.731551\pi$
$541$	−30.9331	−1.32992	−0.664960	+	0.746879i	$0.731551\pi$
$542$	−72.5342	−3.11561
$543$	0	0
$544$	−0.757577	−0.0324808
$545$	17.9253	0.767837
$546$	0	0
$547$	−7.94501	−0.339704	−0.169852	−	0.985470i	$-0.554329\pi$
$547$	−7.94501	−0.339704	−0.169852	+	0.985470i	$0.554329\pi$
$548$	−6.33406	−0.270578
$549$	0	0
$550$	−21.6653	−0.923810
$551$	3.00985	0.128224
$552$	0	0
$553$	21.0472	0.895018
$554$	−45.9414	−1.95186
$555$	0	0
$556$	54.7495	2.32189
$557$	38.1274	1.61551	0.807754	−	0.589519i	$-0.200683\pi$
$557$	38.1274	1.61551	0.807754	+	0.589519i	$0.200683\pi$
$558$	0	0
$559$	20.5087	0.867425
$560$	11.9607	0.505432
$561$	0	0
$562$	48.1090	2.02936
$563$	20.7635	0.875076	0.437538	−	0.899200i	$-0.355850\pi$
$563$	20.7635	0.875076	0.437538	+	0.899200i	$0.355850\pi$
$564$	0	0
$565$	−14.1798	−0.596549
$566$	−6.34855	−0.266849
$567$	0	0
$568$	−12.8076	−0.537394
$569$	−20.6258	−0.864678	−0.432339	−	0.901711i	$-0.642312\pi$
$569$	−20.6258	−0.864678	−0.432339	+	0.901711i	$0.642312\pi$
$570$	0	0
$571$	−2.18892	−0.0916035	−0.0458018	−	0.998951i	$-0.514584\pi$
$571$	−2.18892	−0.0916035	−0.0458018	+	0.998951i	$0.514584\pi$
$572$	53.2493	2.22647
$573$	0	0
$574$	−27.2154	−1.13595
$575$	10.1653	0.423921
$576$	0	0
$577$	12.2941	0.511809	0.255904	−	0.966702i	$-0.417627\pi$
$577$	12.2941	0.511809	0.255904	+	0.966702i	$0.417627\pi$
$578$	−33.3233	−1.38607
$579$	0	0
$580$	12.0162	0.498946
$581$	26.3376	1.09267
$582$	0	0
$583$	1.79970	0.0745361
$584$	67.0164	2.77316
$585$	0	0
$586$	62.1290	2.56653
$587$	−38.3831	−1.58424	−0.792120	−	0.610366i	$-0.791022\pi$
$587$	−38.3831	−1.58424	−0.792120	+	0.610366i	$0.791022\pi$
$588$	0	0
$589$	4.37755	0.180374
$590$	3.09661	0.127485
$591$	0	0
$592$	15.3909	0.632563
$593$	22.9646	0.943041	0.471521	−	0.881855i	$-0.343705\pi$
$593$	22.9646	0.943041	0.471521	+	0.881855i	$0.343705\pi$
$594$	0	0
$595$	16.1988	0.664087
$596$	−94.6209	−3.87582
$597$	0	0
$598$	−37.4333	−1.53076
$599$	14.1726	0.579078	0.289539	−	0.957166i	$-0.406498\pi$
$599$	14.1726	0.579078	0.289539	+	0.957166i	$0.406498\pi$
$600$	0	0
$601$	−7.47443	−0.304888	−0.152444	−	0.988312i	$-0.548714\pi$
$601$	−7.47443	−0.304888	−0.152444	+	0.988312i	$0.548714\pi$
$602$	−24.5913	−1.00227
$603$	0	0
$604$	−5.48041	−0.222995
$605$	6.11488	0.248605
$606$	0	0
$607$	41.2464	1.67414	0.837069	−	0.547097i	$-0.184267\pi$
$607$	41.2464	1.67414	0.837069	+	0.547097i	$0.184267\pi$
$608$	0.136977	0.00555516
$609$	0	0
$610$	−12.6013	−0.510212
$611$	−6.02261	−0.243649
$612$	0	0
$613$	6.42285	0.259417	0.129708	−	0.991552i	$-0.458596\pi$
$613$	6.42285	0.259417	0.129708	+	0.991552i	$0.458596\pi$
$614$	65.5601	2.64579
$615$	0	0
$616$	−32.0354	−1.29074
$617$	−16.3909	−0.659873	−0.329937	−	0.944003i	$-0.607027\pi$
$617$	−16.3909	−0.659873	−0.329937	+	0.944003i	$0.607027\pi$
$618$	0	0
$619$	−30.0059	−1.20604	−0.603020	−	0.797726i	$-0.706036\pi$
$619$	−30.0059	−1.20604	−0.603020	+	0.797726i	$0.706036\pi$
$620$	17.4765	0.701873
$621$	0	0
$622$	−73.2186	−2.93580
$623$	27.2354	1.09116
$624$	0	0
$625$	11.1397	0.445589
$626$	64.2315	2.56721
$627$	0	0
$628$	−23.2465	−0.927638
$629$	20.8445	0.831123
$630$	0	0
$631$	−23.5227	−0.936422	−0.468211	−	0.883617i	$-0.655101\pi$
$631$	−23.5227	−0.936422	−0.468211	+	0.883617i	$0.655101\pi$
$632$	35.2992	1.40413
$633$	0	0
$634$	−17.6049	−0.699181
$635$	7.79393	0.309293
$636$	0	0
$637$	−10.0673	−0.398881
$638$	−16.2586	−0.643685
$639$	0	0
$640$	−19.3744	−0.765840
$641$	19.3846	0.765647	0.382824	−	0.923821i	$-0.374952\pi$
$641$	19.3846	0.765647	0.382824	+	0.923821i	$0.374952\pi$
$642$	0	0
$643$	−28.6706	−1.13066	−0.565328	−	0.824866i	$-0.691251\pi$
$643$	−28.6706	−1.13066	−0.565328	+	0.824866i	$0.691251\pi$
$644$	29.9580	1.18051
$645$	0	0
$646$	13.5630	0.533631
$647$	41.6600	1.63782	0.818912	−	0.573919i	$-0.194578\pi$
$647$	41.6600	1.63782	0.818912	+	0.573919i	$0.194578\pi$
$648$	0	0
$649$	−2.79649	−0.109772
$650$	−59.2363	−2.32344
$651$	0	0
$652$	48.3436	1.89328
$653$	−11.6872	−0.457355	−0.228677	−	0.973502i	$-0.573440\pi$
$653$	−11.6872	−0.457355	−0.228677	+	0.973502i	$0.573440\pi$
$654$	0	0
$655$	12.6611	0.494709
$656$	−15.3900	−0.600880
$657$	0	0
$658$	7.22151	0.281524
$659$	−45.9612	−1.79040	−0.895198	−	0.445669i	$-0.852966\pi$
$659$	−45.9612	−1.79040	−0.895198	+	0.445669i	$0.852966\pi$
$660$	0	0
$661$	−3.41850	−0.132964	−0.0664820	−	0.997788i	$-0.521178\pi$
$661$	−3.41850	−0.132964	−0.0664820	+	0.997788i	$0.521178\pi$
$662$	7.20295	0.279951
$663$	0	0
$664$	44.1720	1.71421
$665$	−2.92890	−0.113578
$666$	0	0
$667$	7.62850	0.295377
$668$	−28.3563	−1.09714
$669$	0	0
$670$	−4.27870	−0.165301
$671$	11.3800	0.439320
$672$	0	0
$673$	32.2185	1.24193	0.620966	−	0.783838i	$-0.286741\pi$
$673$	32.2185	1.24193	0.620966	+	0.783838i	$0.286741\pi$
$674$	26.3082	1.01336
$675$	0	0
$676$	93.4113	3.59274
$677$	−21.0401	−0.808636	−0.404318	−	0.914619i	$-0.632491\pi$
$677$	−21.0401	−0.808636	−0.404318	+	0.914619i	$0.632491\pi$
$678$	0	0
$679$	8.77085	0.336594
$680$	27.1678	1.04184
$681$	0	0
$682$	−23.6467	−0.905479
$683$	22.0959	0.845478	0.422739	−	0.906252i	$-0.361069\pi$
$683$	22.0959	0.845478	0.422739	+	0.906252i	$0.361069\pi$
$684$	0	0
$685$	1.56953	0.0599687
$686$	−38.4792	−1.46914
$687$	0	0
$688$	−13.9061	−0.530165
$689$	4.92068	0.187463
$690$	0	0
$691$	18.8308	0.716358	0.358179	−	0.933653i	$-0.383398\pi$
$691$	18.8308	0.716358	0.358179	+	0.933653i	$0.383398\pi$
$692$	−5.26534	−0.200158
$693$	0	0
$694$	11.7896	0.447527
$695$	−13.5665	−0.514606
$696$	0	0
$697$	−20.8433	−0.789496
$698$	78.2729	2.96267
$699$	0	0
$700$	47.4069	1.79181
$701$	14.3830	0.543238	0.271619	−	0.962405i	$-0.412441\pi$
$701$	14.3830	0.543238	0.271619	+	0.962405i	$0.412441\pi$
$702$	0	0
$703$	−3.76888	−0.142146
$704$	17.2506	0.650156
$705$	0	0
$706$	−49.2729	−1.85441
$707$	9.09813	0.342170
$708$	0	0
$709$	1.07850	0.0405038	0.0202519	−	0.999795i	$-0.493553\pi$
$709$	1.07850	0.0405038	0.0202519	+	0.999795i	$0.493553\pi$
$710$	6.32530	0.237384
$711$	0	0
$712$	45.6777	1.71184
$713$	11.0950	0.415509
$714$	0	0
$715$	−13.1948	−0.493456
$716$	−10.6324	−0.397353
$717$	0	0
$718$	73.8944	2.75772
$719$	−2.79824	−0.104357	−0.0521783	−	0.998638i	$-0.516616\pi$
$719$	−2.79824	−0.104357	−0.0521783	+	0.998638i	$0.516616\pi$
$720$	0	0
$721$	39.7493	1.48034
$722$	−2.45233	−0.0912662
$723$	0	0
$724$	80.5927	2.99520
$725$	12.0717	0.448332
$726$	0	0
$727$	7.84404	0.290919	0.145460	−	0.989364i	$-0.453534\pi$
$727$	7.84404	0.290919	0.145460	+	0.989364i	$0.453534\pi$
$728$	−87.5899	−3.24630
$729$	0	0
$730$	−33.0975	−1.22499
$731$	−18.8335	−0.696583
$732$	0	0
$733$	−46.1833	−1.70582	−0.852910	−	0.522059i	$-0.825164\pi$
$733$	−46.1833	−1.70582	−0.852910	+	0.522059i	$0.825164\pi$
$734$	10.4818	0.386890
$735$	0	0
$736$	0.347170	0.0127969
$737$	3.86401	0.142333
$738$	0	0
$739$	26.8325	0.987049	0.493525	−	0.869732i	$-0.335708\pi$
$739$	26.8325	0.987049	0.493525	+	0.869732i	$0.335708\pi$
$740$	−15.0465	−0.553121
$741$	0	0
$742$	−5.90022	−0.216604
$743$	5.12580	0.188047	0.0940237	−	0.995570i	$-0.470027\pi$
$743$	5.12580	0.188047	0.0940237	+	0.995570i	$0.470027\pi$
$744$	0	0
$745$	23.4463	0.859006
$746$	−65.6207	−2.40254
$747$	0	0
$748$	−48.8999	−1.78796
$749$	4.96433	0.181393
$750$	0	0
$751$	−37.1494	−1.35560	−0.677801	−	0.735246i	$-0.737067\pi$
$751$	−37.1494	−1.35560	−0.677801	+	0.735246i	$0.737067\pi$
$752$	4.08369	0.148917
$753$	0	0
$754$	−44.4537	−1.61891
$755$	1.35800	0.0494227
$756$	0	0
$757$	38.6892	1.40618	0.703091	−	0.711100i	$-0.251803\pi$
$757$	38.6892	1.40618	0.703091	+	0.711100i	$0.251803\pi$
$758$	−29.3605	−1.06642
$759$	0	0
$760$	−4.91219	−0.178184
$761$	−19.0671	−0.691183	−0.345592	−	0.938385i	$-0.612322\pi$
$761$	−19.0671	−0.691183	−0.345592	+	0.938385i	$0.612322\pi$
$762$	0	0
$763$	53.0715	1.92132
$764$	−89.7084	−3.24554
$765$	0	0
$766$	−14.0192	−0.506534
$767$	−7.64606	−0.276083
$768$	0	0
$769$	−42.6273	−1.53718	−0.768589	−	0.639742i	$-0.779041\pi$
$769$	−42.6273	−1.53718	−0.768589	+	0.639742i	$0.779041\pi$
$770$	15.8214	0.570163
$771$	0	0
$772$	33.1893	1.19451
$773$	−28.9259	−1.04039	−0.520197	−	0.854046i	$-0.674141\pi$
$773$	−28.9259	−1.04039	−0.520197	+	0.854046i	$0.674141\pi$
$774$	0	0
$775$	17.5572	0.630673
$776$	14.7100	0.528058
$777$	0	0
$778$	83.3811	2.98936
$779$	3.76866	0.135026
$780$	0	0
$781$	−5.71226	−0.204401
$782$	34.3757	1.22927
$783$	0	0
$784$	6.82622	0.243794
$785$	5.76030	0.205594
$786$	0	0
$787$	21.1198	0.752840	0.376420	−	0.926449i	$-0.377155\pi$
$787$	21.1198	0.752840	0.376420	+	0.926449i	$0.377155\pi$
$788$	−24.7071	−0.880153
$789$	0	0
$790$	−17.4333	−0.620249
$791$	−41.9821	−1.49271
$792$	0	0
$793$	31.1148	1.10492
$794$	−36.5814	−1.29822
$795$	0	0
$796$	−57.4783	−2.03727
$797$	37.9370	1.34380	0.671900	−	0.740642i	$-0.265479\pi$
$797$	37.9370	1.34380	0.671900	+	0.740642i	$0.265479\pi$
$798$	0	0
$799$	5.53068	0.195661
$800$	0.549379	0.0194235
$801$	0	0
$802$	16.4053	0.579292
$803$	29.8898	1.05479
$804$	0	0
$805$	−7.42334	−0.261638
$806$	−64.6538	−2.27733
$807$	0	0
$808$	15.2589	0.536805
$809$	5.16698	0.181661	0.0908307	−	0.995866i	$-0.471048\pi$
$809$	5.16698	0.181661	0.0908307	+	0.995866i	$0.471048\pi$
$810$	0	0
$811$	13.7373	0.482383	0.241191	−	0.970478i	$-0.422462\pi$
$811$	13.7373	0.482383	0.241191	+	0.970478i	$0.422462\pi$
$812$	35.5764	1.24849
$813$	0	0
$814$	20.3588	0.713575
$815$	−11.9792	−0.419612
$816$	0	0
$817$	3.40528	0.119136
$818$	23.5612	0.823800
$819$	0	0
$820$	15.0457	0.525417
$821$	18.5531	0.647506	0.323753	−	0.946142i	$-0.395055\pi$
$821$	18.5531	0.647506	0.323753	+	0.946142i	$0.395055\pi$
$822$	0	0
$823$	−5.44721	−0.189878	−0.0949388	−	0.995483i	$-0.530266\pi$
$823$	−5.44721	−0.189878	−0.0949388	+	0.995483i	$0.530266\pi$
$824$	66.6654	2.32240
$825$	0	0
$826$	9.16813	0.319000
$827$	−44.5724	−1.54993	−0.774966	−	0.632003i	$-0.782233\pi$
$827$	−44.5724	−1.54993	−0.774966	+	0.632003i	$0.782233\pi$
$828$	0	0
$829$	39.4430	1.36991	0.684955	−	0.728585i	$-0.259822\pi$
$829$	39.4430	1.36991	0.684955	+	0.728585i	$0.259822\pi$
$830$	−21.8153	−0.757221
$831$	0	0
$832$	47.1659	1.63518
$833$	9.24500	0.320320
$834$	0	0
$835$	7.02645	0.243160
$836$	8.84157	0.305792
$837$	0	0
$838$	−10.8009	−0.373112
$839$	8.64856	0.298582	0.149291	−	0.988793i	$-0.452301\pi$
$839$	8.64856	0.298582	0.149291	+	0.988793i	$0.452301\pi$
$840$	0	0
$841$	−19.9408	−0.687615
$842$	−10.2426	−0.352985
$843$	0	0
$844$	62.2214	2.14175
$845$	−23.1466	−0.796266
$846$	0	0
$847$	18.1043	0.622072
$848$	−3.33651	−0.114576
$849$	0	0
$850$	54.3979	1.86583
$851$	−9.55228	−0.327448
$852$	0	0
$853$	41.0903	1.40691	0.703453	−	0.710742i	$-0.251641\pi$
$853$	41.0903	1.40691	0.703453	+	0.710742i	$0.251641\pi$
$854$	−37.3086	−1.27668
$855$	0	0
$856$	8.32590	0.284573
$857$	−10.2507	−0.350158	−0.175079	−	0.984554i	$-0.556018\pi$
$857$	−10.2507	−0.350158	−0.175079	+	0.984554i	$0.556018\pi$
$858$	0	0
$859$	18.5872	0.634186	0.317093	−	0.948394i	$-0.397293\pi$
$859$	18.5872	0.634186	0.317093	+	0.948394i	$0.397293\pi$
$860$	13.5949	0.463583
$861$	0	0
$862$	8.36005	0.284744
$863$	−7.01800	−0.238896	−0.119448	−	0.992840i	$-0.538112\pi$
$863$	−7.01800	−0.238896	−0.119448	+	0.992840i	$0.538112\pi$
$864$	0	0
$865$	1.30471	0.0443615
$866$	−25.4972	−0.866429
$867$	0	0
$868$	51.7426	1.75626
$869$	15.7437	0.534068
$870$	0	0
$871$	10.5648	0.357976
$872$	89.0086	3.01421
$873$	0	0
$874$	−6.21546	−0.210241
$875$	−26.3916	−0.892197
$876$	0	0
$877$	50.0869	1.69131	0.845657	−	0.533726i	$-0.179209\pi$
$877$	50.0869	1.69131	0.845657	+	0.533726i	$0.179209\pi$
$878$	−20.4877	−0.691426
$879$	0	0
$880$	8.94682	0.301597
$881$	5.49731	0.185209	0.0926045	−	0.995703i	$-0.470481\pi$
$881$	5.49731	0.185209	0.0926045	+	0.995703i	$0.470481\pi$
$882$	0	0
$883$	7.23660	0.243531	0.121765	−	0.992559i	$-0.461144\pi$
$883$	7.23660	0.243531	0.121765	+	0.992559i	$0.461144\pi$
$884$	−133.700	−4.49682
$885$	0	0
$886$	−27.1044	−0.910590
$887$	48.1420	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$887$	48.1420	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$888$	0	0
$889$	23.0754	0.773926
$890$	−22.5589	−0.756177
$891$	0	0
$892$	−17.3949	−0.582425
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	2.63463	0.0880660
$896$	−57.3617	−1.91632
$897$	0	0
$898$	9.42252	0.314433
$899$	13.1757	0.439436
$900$	0	0
$901$	−4.51876	−0.150542
$902$	−20.3576	−0.677835
$903$	0	0
$904$	−70.4100	−2.34180
$905$	−19.9702	−0.663832
$906$	0	0
$907$	−17.4917	−0.580802	−0.290401	−	0.956905i	$-0.593789\pi$
$907$	−17.4917	−0.580802	−0.290401	+	0.956905i	$0.593789\pi$
$908$	92.2220	3.06050
$909$	0	0
$910$	43.2582	1.43399
$911$	35.8314	1.18715	0.593573	−	0.804780i	$-0.297717\pi$
$911$	35.8314	1.18715	0.593573	+	0.804780i	$0.297717\pi$
$912$	0	0
$913$	19.7010	0.652008
$914$	80.1738	2.65191
$915$	0	0
$916$	−70.9347	−2.34375
$917$	37.4856	1.23788
$918$	0	0
$919$	−29.5752	−0.975596	−0.487798	−	0.872956i	$-0.662200\pi$
$919$	−29.5752	−0.975596	−0.487798	+	0.872956i	$0.662200\pi$
$920$	−12.4500	−0.410465
$921$	0	0
$922$	−8.05435	−0.265256
$923$	−15.6182	−0.514080
$924$	0	0
$925$	−15.1160	−0.497011
$926$	34.9567	1.14875
$927$	0	0
$928$	0.412280	0.0135338
$929$	−32.4330	−1.06409	−0.532046	−	0.846715i	$-0.678577\pi$
$929$	−32.4330	−1.06409	−0.532046	+	0.846715i	$0.678577\pi$
$930$	0	0
$931$	−1.67158	−0.0547839
$932$	65.2860	2.13851
$933$	0	0
$934$	35.0502	1.14688
$935$	12.1170	0.396268
$936$	0	0
$937$	46.7864	1.52845	0.764223	−	0.644952i	$-0.223123\pi$
$937$	46.7864	1.52845	0.764223	+	0.644952i	$0.223123\pi$
$938$	−12.6679	−0.413622
$939$	0	0
$940$	−3.99230	−0.130215
$941$	24.1501	0.787272	0.393636	−	0.919266i	$-0.371217\pi$
$941$	24.1501	0.787272	0.393636	+	0.919266i	$0.371217\pi$
$942$	0	0
$943$	9.55173	0.311047
$944$	5.18448	0.168740
$945$	0	0
$946$	−18.3947	−0.598063
$947$	16.5765	0.538665	0.269333	−	0.963047i	$-0.413197\pi$
$947$	16.5765	0.538665	0.269333	+	0.963047i	$0.413197\pi$
$948$	0	0
$949$	81.7234	2.65285
$950$	−9.83565	−0.319111
$951$	0	0
$952$	80.4355	2.60693
$953$	0.237569	0.00769561	0.00384780	−	0.999993i	$-0.498775\pi$
$953$	0.237569	0.00769561	0.00384780	+	0.999993i	$0.498775\pi$
$954$	0	0
$955$	22.2290	0.719314
$956$	9.93258	0.321243
$957$	0	0
$958$	−94.4949	−3.05299
$959$	4.64690	0.150056
$960$	0	0
$961$	−11.8371	−0.381842
$962$	55.6642	1.79468
$963$	0	0
$964$	−11.5257	−0.371219
$965$	−8.22404	−0.264741
$966$	0	0
$967$	26.8566	0.863648	0.431824	−	0.901958i	$-0.357870\pi$
$967$	26.8566	0.863648	0.431824	+	0.901958i	$0.357870\pi$
$968$	30.3636	0.975922
$969$	0	0
$970$	−7.26485	−0.233260
$971$	52.1921	1.67493	0.837463	−	0.546495i	$-0.184038\pi$
$971$	52.1921	1.67493	0.837463	+	0.546495i	$0.184038\pi$
$972$	0	0
$973$	−40.1662	−1.28767
$974$	35.9516	1.15196
$975$	0	0
$976$	−21.0976	−0.675319
$977$	15.4893	0.495546	0.247773	−	0.968818i	$-0.420301\pi$
$977$	15.4893	0.495546	0.247773	+	0.968818i	$0.420301\pi$
$978$	0	0
$979$	20.3726	0.651110
$980$	−6.67347	−0.213176
$981$	0	0
$982$	−14.5287	−0.463629
$983$	57.7195	1.84097	0.920483	−	0.390782i	$-0.127795\pi$
$983$	57.7195	1.84097	0.920483	+	0.390782i	$0.127795\pi$
$984$	0	0
$985$	6.12221	0.195070
$986$	40.8227	1.30006
$987$	0	0
$988$	24.1743	0.769086
$989$	8.63074	0.274441
$990$	0	0
$991$	−38.9376	−1.23689	−0.618446	−	0.785827i	$-0.712238\pi$
$991$	−38.9376	−1.23689	−0.618446	+	0.785827i	$0.712238\pi$
$992$	0.599624	0.0190381
$993$	0	0
$994$	18.7273	0.593993
$995$	14.2427	0.451523
$996$	0	0
$997$	−13.1366	−0.416039	−0.208020	−	0.978125i	$-0.566702\pi$
$997$	−13.1366	−0.416039	−0.208020	+	0.978125i	$0.566702\pi$
$998$	−28.9800	−0.917346
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.w.1.2	yes	34
3.2	odd	2		8037.2.a.v.1.33	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.v.1.33	✓	34	3.2	odd	2
8037.2.a.w.1.2	yes	34	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)

\(f(q)\)	\(=\)	\(q-2.45233 q^{2} +4.01392 q^{4} -0.994616 q^{5} -2.94476 q^{7} -4.93878 q^{8} +O(q^{10})\)	\(q-2.45233 q^{2} +4.01392 q^{4} -0.994616 q^{5} -2.94476 q^{7} -4.93878 q^{8} +2.43912 q^{10} -2.20273 q^{11} -6.02261 q^{13} +7.22151 q^{14} +4.08369 q^{16} +5.53068 q^{17} -1.00000 q^{19} -3.99230 q^{20} +5.40181 q^{22} -2.53451 q^{23} -4.01074 q^{25} +14.7694 q^{26} -11.8200 q^{28} -3.00985 q^{29} -4.37755 q^{31} -0.136977 q^{32} -13.5630 q^{34} +2.92890 q^{35} +3.76888 q^{37} +2.45233 q^{38} +4.91219 q^{40} -3.76866 q^{41} -3.40528 q^{43} -8.84157 q^{44} +6.21546 q^{46} +1.00000 q^{47} +1.67158 q^{49} +9.83565 q^{50} -24.1743 q^{52} -0.817034 q^{53} +2.19087 q^{55} +14.5435 q^{56} +7.38113 q^{58} +1.26956 q^{59} -5.16632 q^{61} +10.7352 q^{62} -7.83146 q^{64} +5.99019 q^{65} -1.75419 q^{67} +22.1997 q^{68} -7.18263 q^{70} +2.59327 q^{71} -13.5694 q^{73} -9.24253 q^{74} -4.01392 q^{76} +6.48650 q^{77} -7.14735 q^{79} -4.06170 q^{80} +9.24200 q^{82} -8.94391 q^{83} -5.50090 q^{85} +8.35087 q^{86} +10.8788 q^{88} -9.24878 q^{89} +17.7351 q^{91} -10.1733 q^{92} -2.45233 q^{94} +0.994616 q^{95} -2.97847 q^{97} -4.09927 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * q^97 + 45 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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