LMFDB - Embedded newform 8037.2.a.k.1.1

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 9x^{5} + 14x^{4} + 23x^{3} - 19x^{2} - 14x + 5 $ x^7 - 2x^6 - 9x^5 + 14x^4 + 23x^3 - 19x^2 - 14x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2679)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.26927$ 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)

$f(q)$	$=$	$q-2.26927 q^{2} +3.14959 q^{4} +3.56052 q^{5} -0.858342 q^{7} -2.60873 q^{8} +O(q^{10})$	\(q-2.26927 q^{2} +3.14959 q^{4} +3.56052 q^{5} -0.858342 q^{7} -2.60873 q^{8} -8.07978 q^{10} +2.00968 q^{11} -1.96834 q^{13} +1.94781 q^{14} -0.379268 q^{16} +0.0701869 q^{17} -1.00000 q^{19} +11.2142 q^{20} -4.56052 q^{22} +5.00000 q^{23} +7.67728 q^{25} +4.46669 q^{26} -2.70342 q^{28} +2.03166 q^{29} +1.13999 q^{31} +6.07812 q^{32} -0.159273 q^{34} -3.05614 q^{35} +5.52353 q^{37} +2.26927 q^{38} -9.28842 q^{40} +4.32146 q^{41} -7.78728 q^{43} +6.32968 q^{44} -11.3464 q^{46} -1.00000 q^{47} -6.26325 q^{49} -17.4218 q^{50} -6.19946 q^{52} +8.40955 q^{53} +7.15552 q^{55} +2.23918 q^{56} -4.61039 q^{58} +9.34206 q^{59} -0.845082 q^{61} -2.58695 q^{62} -13.0344 q^{64} -7.00831 q^{65} +2.96060 q^{67} +0.221060 q^{68} +6.93521 q^{70} +1.04686 q^{71} -0.110033 q^{73} -12.5344 q^{74} -3.14959 q^{76} -1.72500 q^{77} +6.69111 q^{79} -1.35039 q^{80} -9.80656 q^{82} -8.38322 q^{83} +0.249902 q^{85} +17.6715 q^{86} -5.24272 q^{88} -10.3364 q^{89} +1.68951 q^{91} +15.7479 q^{92} +2.26927 q^{94} -3.56052 q^{95} -4.28407 q^{97} +14.2130 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 8 q^{4} + 6 q^{5} + 7 q^{7} + 12 q^{8}+O(q^{10}) $ 7 * q + 2 * q^2 + 8 * q^4 + 6 * q^5 + 7 * q^7 + 12 * q^8	$ 7 q + 2 q^{2} + 8 q^{4} + 6 q^{5} + 7 q^{7} + 12 q^{8} - 2 q^{10} + 9 q^{11} - 2 q^{13} + 6 q^{14} + 14 q^{16} + 6 q^{17} - 7 q^{19} + 25 q^{20} - 13 q^{22} + 35 q^{23} + 3 q^{25} + 23 q^{26} - 9 q^{28} + 26 q^{29} - 11 q^{31} + 21 q^{32} + 18 q^{34} + 17 q^{35} - 3 q^{37} - 2 q^{38} + 14 q^{40} + 20 q^{41} - 5 q^{43} - 18 q^{44} + 10 q^{46} - 7 q^{47} - 10 q^{49} - 17 q^{50} + 15 q^{52} - 3 q^{53} - 3 q^{55} + 3 q^{56} + 31 q^{58} + 15 q^{59} - 9 q^{61} + 4 q^{62} + 4 q^{64} + 13 q^{65} - 4 q^{67} - 5 q^{68} + 18 q^{70} + 4 q^{71} - 8 q^{73} - 28 q^{74} - 8 q^{76} + 11 q^{77} - 11 q^{79} + 12 q^{80} - 19 q^{82} + 4 q^{83} - 6 q^{85} + 36 q^{86} - 7 q^{88} + 23 q^{89} + 19 q^{91} + 40 q^{92} - 2 q^{94} - 6 q^{95} - 7 q^{97}+O(q^{100}) $ 7 * q + 2 * q^2 + 8 * q^4 + 6 * q^5 + 7 * q^7 + 12 * q^8 - 2 * q^10 + 9 * q^11 - 2 * q^13 + 6 * q^14 + 14 * q^16 + 6 * q^17 - 7 * q^19 + 25 * q^20 - 13 * q^22 + 35 * q^23 + 3 * q^25 + 23 * q^26 - 9 * q^28 + 26 * q^29 - 11 * q^31 + 21 * q^32 + 18 * q^34 + 17 * q^35 - 3 * q^37 - 2 * q^38 + 14 * q^40 + 20 * q^41 - 5 * q^43 - 18 * q^44 + 10 * q^46 - 7 * q^47 - 10 * q^49 - 17 * q^50 + 15 * q^52 - 3 * q^53 - 3 * q^55 + 3 * q^56 + 31 * q^58 + 15 * q^59 - 9 * q^61 + 4 * q^62 + 4 * q^64 + 13 * q^65 - 4 * q^67 - 5 * q^68 + 18 * q^70 + 4 * q^71 - 8 * q^73 - 28 * q^74 - 8 * q^76 + 11 * q^77 - 11 * q^79 + 12 * q^80 - 19 * q^82 + 4 * q^83 - 6 * q^85 + 36 * q^86 - 7 * q^88 + 23 * q^89 + 19 * q^91 + 40 * q^92 - 2 * q^94 - 6 * q^95 - 7 * q^97

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 $

$2$	−2.26927	−1.60462	−0.802308	−	0.596910i	$-0.796395\pi$
$2$	−2.26927	−1.60462	−0.802308	+	0.596910i	$0.796395\pi$
$3$	0	0
$3$	0	0
$4$	3.14959	1.57479
$5$	3.56052	1.59231	0.796156	−	0.605092i	$-0.206863\pi$
$5$	3.56052	1.59231	0.796156	+	0.605092i	$0.206863\pi$
$6$	0	0
$7$	−0.858342	−0.324423	−0.162211	−	0.986756i	$-0.551863\pi$
$7$	−0.858342	−0.324423	−0.162211	+	0.986756i	$0.551863\pi$
$8$	−2.60873	−0.922325
$9$	0	0
$10$	−8.07978	−2.55505
$11$	2.00968	0.605943	0.302971	−	0.953000i	$-0.402021\pi$
$11$	2.00968	0.605943	0.302971	+	0.953000i	$0.402021\pi$
$12$	0	0
$13$	−1.96834	−0.545919	−0.272960	−	0.962026i	$-0.588003\pi$
$13$	−1.96834	−0.545919	−0.272960	+	0.962026i	$0.588003\pi$
$14$	1.94781	0.520574
$15$	0	0
$16$	−0.379268	−0.0948170
$17$	0.0701869	0.0170228	0.00851142	−	0.999964i	$-0.497291\pi$
$17$	0.0701869	0.0170228	0.00851142	+	0.999964i	$0.497291\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	11.2142	2.50756
$21$	0	0
$22$	−4.56052	−0.972306
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	0	0
$25$	7.67728	1.53546
$26$	4.46669	0.875991
$27$	0	0
$28$	−2.70342	−0.510899
$29$	2.03166	0.377270	0.188635	−	0.982047i	$-0.439594\pi$
$29$	2.03166	0.377270	0.188635	+	0.982047i	$0.439594\pi$
$30$	0	0
$31$	1.13999	0.204749	0.102374	−	0.994746i	$-0.467356\pi$
$31$	1.13999	0.204749	0.102374	+	0.994746i	$0.467356\pi$
$32$	6.07812	1.07447
$33$	0	0
$34$	−0.159273	−0.0273151
$35$	−3.05614	−0.516582
$36$	0	0
$37$	5.52353	0.908062	0.454031	−	0.890986i	$-0.349985\pi$
$37$	5.52353	0.908062	0.454031	+	0.890986i	$0.349985\pi$
$38$	2.26927	0.368124
$39$	0	0
$40$	−9.28842	−1.46863
$41$	4.32146	0.674899	0.337449	−	0.941344i	$-0.390436\pi$
$41$	4.32146	0.674899	0.337449	+	0.941344i	$0.390436\pi$
$42$	0	0
$43$	−7.78728	−1.18755	−0.593775	−	0.804631i	$-0.702363\pi$
$43$	−7.78728	−1.18755	−0.593775	+	0.804631i	$0.702363\pi$
$44$	6.32968	0.954235
$45$	0	0
$46$	−11.3464	−1.67293
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−6.26325	−0.894750
$50$	−17.4218	−2.46382
$51$	0	0
$52$	−6.19946	−0.859710
$53$	8.40955	1.15514	0.577570	−	0.816341i	$-0.304001\pi$
$53$	8.40955	1.15514	0.577570	+	0.816341i	$0.304001\pi$
$54$	0	0
$55$	7.15552	0.964850
$56$	2.23918	0.299223
$57$	0	0
$58$	−4.61039	−0.605374
$59$	9.34206	1.21623	0.608116	−	0.793848i	$-0.291925\pi$
$59$	9.34206	1.21623	0.608116	+	0.793848i	$0.291925\pi$
$60$	0	0
$61$	−0.845082	−0.108202	−0.0541008	−	0.998535i	$-0.517229\pi$
$61$	−0.845082	−0.108202	−0.0541008	+	0.998535i	$0.517229\pi$
$62$	−2.58695	−0.328543
$63$	0	0
$64$	−13.0344	−1.62929
$65$	−7.00831	−0.869273
$66$	0	0
$67$	2.96060	0.361695	0.180848	−	0.983511i	$-0.442116\pi$
$67$	2.96060	0.361695	0.180848	+	0.983511i	$0.442116\pi$
$68$	0.221060	0.0268075
$69$	0	0
$70$	6.93521	0.828916
$71$	1.04686	0.124240	0.0621198	−	0.998069i	$-0.480214\pi$
$71$	1.04686	0.124240	0.0621198	+	0.998069i	$0.480214\pi$
$72$	0	0
$73$	−0.110033	−0.0128784	−0.00643922	−	0.999979i	$-0.502050\pi$
$73$	−0.110033	−0.0128784	−0.00643922	+	0.999979i	$0.502050\pi$
$74$	−12.5344	−1.45709
$75$	0	0
$76$	−3.14959	−0.361283
$77$	−1.72500	−0.196582
$78$	0	0
$79$	6.69111	0.752809	0.376405	−	0.926455i	$-0.377160\pi$
$79$	6.69111	0.752809	0.376405	+	0.926455i	$0.377160\pi$
$80$	−1.35039	−0.150978
$81$	0	0
$82$	−9.80656	−1.08295
$83$	−8.38322	−0.920178	−0.460089	−	0.887873i	$-0.652182\pi$
$83$	−8.38322	−0.920178	−0.460089	+	0.887873i	$0.652182\pi$
$84$	0	0
$85$	0.249902	0.0271057
$86$	17.6715	1.90556
$87$	0	0
$88$	−5.24272	−0.558876
$89$	−10.3364	−1.09566	−0.547831	−	0.836589i	$-0.684546\pi$
$89$	−10.3364	−1.09566	−0.547831	+	0.836589i	$0.684546\pi$
$90$	0	0
$91$	1.68951	0.177109
$92$	15.7479	1.64184
$93$	0	0
$94$	2.26927	0.234057
$95$	−3.56052	−0.365301
$96$	0	0
$97$	−4.28407	−0.434981	−0.217490	−	0.976062i	$-0.569787\pi$
$97$	−4.28407	−0.434981	−0.217490	+	0.976062i	$0.569787\pi$
$98$	14.2130	1.43573
$99$	0	0
$100$	24.1803	2.41803
$101$	−14.5263	−1.44542	−0.722711	−	0.691150i	$-0.757104\pi$
$101$	−14.5263	−1.44542	−0.722711	+	0.691150i	$0.757104\pi$
$102$	0	0
$103$	9.33918	0.920216	0.460108	−	0.887863i	$-0.347811\pi$
$103$	9.33918	0.920216	0.460108	+	0.887863i	$0.347811\pi$
$104$	5.13486	0.503515
$105$	0	0
$106$	−19.0835	−1.85356
$107$	−11.8979	−1.15022	−0.575108	−	0.818077i	$-0.695040\pi$
$107$	−11.8979	−1.15022	−0.575108	+	0.818077i	$0.695040\pi$
$108$	0	0
$109$	2.98865	0.286261	0.143130	−	0.989704i	$-0.454283\pi$
$109$	2.98865	0.286261	0.143130	+	0.989704i	$0.454283\pi$
$110$	−16.2378	−1.54821
$111$	0	0
$112$	0.325542	0.0307608
$113$	9.85847	0.927406	0.463703	−	0.885991i	$-0.346520\pi$
$113$	9.85847	0.927406	0.463703	+	0.885991i	$0.346520\pi$
$114$	0	0
$115$	17.8026	1.66010
$116$	6.39890	0.594123
$117$	0	0
$118$	−21.1997	−1.95159
$119$	−0.0602444	−0.00552259
$120$	0	0
$121$	−6.96117	−0.632834
$122$	1.91772	0.173622
$123$	0	0
$124$	3.59051	0.322437
$125$	9.53252	0.852614
$126$	0	0
$127$	7.08598	0.628779	0.314389	−	0.949294i	$-0.398200\pi$
$127$	7.08598	0.628779	0.314389	+	0.949294i	$0.398200\pi$
$128$	17.4222	1.53992
$129$	0	0
$130$	15.9037	1.39485
$131$	5.95621	0.520396	0.260198	−	0.965555i	$-0.416212\pi$
$131$	5.95621	0.520396	0.260198	+	0.965555i	$0.416212\pi$
$132$	0	0
$133$	0.858342	0.0744277
$134$	−6.71841	−0.580382
$135$	0	0
$136$	−0.183099	−0.0157006
$137$	13.9459	1.19148	0.595738	−	0.803179i	$-0.296860\pi$
$137$	13.9459	1.19148	0.595738	+	0.803179i	$0.296860\pi$
$138$	0	0
$139$	8.00473	0.678952	0.339476	−	0.940615i	$-0.389750\pi$
$139$	8.00473	0.678952	0.339476	+	0.940615i	$0.389750\pi$
$140$	−9.62559	−0.813511
$141$	0	0
$142$	−2.37561	−0.199357
$143$	−3.95574	−0.330796
$144$	0	0
$145$	7.23376	0.600731
$146$	0.249696	0.0206649
$147$	0	0
$148$	17.3968	1.43001
$149$	−3.17586	−0.260177	−0.130088	−	0.991502i	$-0.541526\pi$
$149$	−3.17586	−0.260177	−0.130088	+	0.991502i	$0.541526\pi$
$150$	0	0
$151$	15.3154	1.24635	0.623177	−	0.782081i	$-0.285842\pi$
$151$	15.3154	1.24635	0.623177	+	0.782081i	$0.285842\pi$
$152$	2.60873	0.211596
$153$	0	0
$154$	3.91448	0.315438
$155$	4.05896	0.326024
$156$	0	0
$157$	24.5684	1.96077	0.980387	−	0.197084i	$-0.0631471\pi$
$157$	24.5684	1.96077	0.980387	+	0.197084i	$0.0631471\pi$
$158$	−15.1839	−1.20797
$159$	0	0
$160$	21.6412	1.71089
$161$	−4.29171	−0.338234
$162$	0	0
$163$	−3.45390	−0.270531	−0.135265	−	0.990809i	$-0.543189\pi$
$163$	−3.45390	−0.270531	−0.135265	+	0.990809i	$0.543189\pi$
$164$	13.6108	1.06283
$165$	0	0
$166$	19.0238	1.47653
$167$	18.6557	1.44362	0.721809	−	0.692092i	$-0.243311\pi$
$167$	18.6557	1.44362	0.721809	+	0.692092i	$0.243311\pi$
$168$	0	0
$169$	−9.12564	−0.701972
$170$	−0.567095	−0.0434942
$171$	0	0
$172$	−24.5267	−1.87015
$173$	2.33711	0.177687	0.0888434	−	0.996046i	$-0.471683\pi$
$173$	2.33711	0.177687	0.0888434	+	0.996046i	$0.471683\pi$
$174$	0	0
$175$	−6.58973	−0.498137
$176$	−0.762209	−0.0574537
$177$	0	0
$178$	23.4562	1.75812
$179$	10.9398	0.817676	0.408838	−	0.912607i	$-0.365934\pi$
$179$	10.9398	0.817676	0.408838	+	0.912607i	$0.365934\pi$
$180$	0	0
$181$	12.2530	0.910759	0.455379	−	0.890298i	$-0.349504\pi$
$181$	12.2530	0.910759	0.455379	+	0.890298i	$0.349504\pi$
$182$	−3.83395	−0.284191
$183$	0	0
$184$	−13.0436	−0.961590
$185$	19.6666	1.44592
$186$	0	0
$187$	0.141054	0.0103149
$188$	−3.14959	−0.229707
$189$	0	0
$190$	8.07978	0.586169
$191$	−6.46623	−0.467880	−0.233940	−	0.972251i	$-0.575162\pi$
$191$	−6.46623	−0.467880	−0.233940	+	0.972251i	$0.575162\pi$
$192$	0	0
$193$	−6.40785	−0.461247	−0.230623	−	0.973043i	$-0.574077\pi$
$193$	−6.40785	−0.461247	−0.230623	+	0.973043i	$0.574077\pi$
$194$	9.72170	0.697978
$195$	0	0
$196$	−19.7267	−1.40905
$197$	−16.1541	−1.15094	−0.575468	−	0.817825i	$-0.695180\pi$
$197$	−16.1541	−1.15094	−0.575468	+	0.817825i	$0.695180\pi$
$198$	0	0
$199$	−8.10361	−0.574449	−0.287225	−	0.957863i	$-0.592733\pi$
$199$	−8.10361	−0.574449	−0.287225	+	0.957863i	$0.592733\pi$
$200$	−20.0279	−1.41619
$201$	0	0
$202$	32.9641	2.31935
$203$	−1.74386	−0.122395
$204$	0	0
$205$	15.3866	1.07465
$206$	−21.1931	−1.47659
$207$	0	0
$208$	0.746528	0.0517624
$209$	−2.00968	−0.139013
$210$	0	0
$211$	−7.62192	−0.524715	−0.262357	−	0.964971i	$-0.584500\pi$
$211$	−7.62192	−0.524715	−0.262357	+	0.964971i	$0.584500\pi$
$212$	26.4866	1.81911
$213$	0	0
$214$	26.9996	1.84566
$215$	−27.7268	−1.89095
$216$	0	0
$217$	−0.978503	−0.0664251
$218$	−6.78205	−0.459339
$219$	0	0
$220$	22.5369	1.51944
$221$	−0.138152	−0.00929309
$222$	0	0
$223$	8.49156	0.568637	0.284318	−	0.958730i	$-0.408233\pi$
$223$	8.49156	0.568637	0.284318	+	0.958730i	$0.408233\pi$
$224$	−5.21710	−0.348582
$225$	0	0
$226$	−22.3715	−1.48813
$227$	18.4809	1.22662	0.613310	−	0.789842i	$-0.289837\pi$
$227$	18.4809	1.22662	0.613310	+	0.789842i	$0.289837\pi$
$228$	0	0
$229$	−3.87836	−0.256290	−0.128145	−	0.991755i	$-0.540902\pi$
$229$	−3.87836	−0.256290	−0.128145	+	0.991755i	$0.540902\pi$
$230$	−40.3989	−2.66382
$231$	0	0
$232$	−5.30005	−0.347965
$233$	17.0528	1.11717	0.558583	−	0.829448i	$-0.311345\pi$
$233$	17.0528	1.11717	0.558583	+	0.829448i	$0.311345\pi$
$234$	0	0
$235$	−3.56052	−0.232263
$236$	29.4236	1.91532
$237$	0	0
$238$	0.136711	0.00886165
$239$	23.5262	1.52178	0.760892	−	0.648879i	$-0.224762\pi$
$239$	23.5262	1.52178	0.760892	+	0.648879i	$0.224762\pi$
$240$	0	0
$241$	−22.0162	−1.41819	−0.709095	−	0.705113i	$-0.750896\pi$
$241$	−22.0162	−1.41819	−0.709095	+	0.705113i	$0.750896\pi$
$242$	15.7968	1.01546
$243$	0	0
$244$	−2.66166	−0.170395
$245$	−22.3004	−1.42472
$246$	0	0
$247$	1.96834	0.125242
$248$	−2.97393	−0.188845
$249$	0	0
$250$	−21.6319	−1.36812
$251$	−0.908175	−0.0573235	−0.0286617	−	0.999589i	$-0.509125\pi$
$251$	−0.908175	−0.0573235	−0.0286617	+	0.999589i	$0.509125\pi$
$252$	0	0
$253$	10.0484	0.631739
$254$	−16.0800	−1.00895
$255$	0	0
$256$	−13.4671	−0.841692
$257$	−19.1297	−1.19328	−0.596639	−	0.802509i	$-0.703498\pi$
$257$	−19.1297	−1.19328	−0.596639	+	0.802509i	$0.703498\pi$
$258$	0	0
$259$	−4.74107	−0.294596
$260$	−22.0733	−1.36893
$261$	0	0
$262$	−13.5163	−0.835037
$263$	17.9420	1.10635	0.553174	−	0.833066i	$-0.313416\pi$
$263$	17.9420	1.10635	0.553174	+	0.833066i	$0.313416\pi$
$264$	0	0
$265$	29.9424	1.83934
$266$	−1.94781	−0.119428
$267$	0	0
$268$	9.32468	0.569595
$269$	−4.35795	−0.265709	−0.132854	−	0.991136i	$-0.542414\pi$
$269$	−4.35795	−0.265709	−0.132854	+	0.991136i	$0.542414\pi$
$270$	0	0
$271$	−0.992785	−0.0603074	−0.0301537	−	0.999545i	$-0.509600\pi$
$271$	−0.992785	−0.0603074	−0.0301537	+	0.999545i	$0.509600\pi$
$272$	−0.0266197	−0.00161405
$273$	0	0
$274$	−31.6470	−1.91186
$275$	15.4289	0.930399
$276$	0	0
$277$	−10.2357	−0.615005	−0.307503	−	0.951547i	$-0.599493\pi$
$277$	−10.2357	−0.615005	−0.307503	+	0.951547i	$0.599493\pi$
$278$	−18.1649	−1.08946
$279$	0	0
$280$	7.97264	0.476456
$281$	18.5035	1.10383	0.551914	−	0.833901i	$-0.313898\pi$
$281$	18.5035	1.10383	0.551914	+	0.833901i	$0.313898\pi$
$282$	0	0
$283$	7.54231	0.448344	0.224172	−	0.974550i	$-0.428032\pi$
$283$	7.54231	0.448344	0.224172	+	0.974550i	$0.428032\pi$
$284$	3.29718	0.195652
$285$	0	0
$286$	8.97665	0.530800
$287$	−3.70929	−0.218952
$288$	0	0
$289$	−16.9951	−0.999710
$290$	−16.4154	−0.963944
$291$	0	0
$292$	−0.346560	−0.0202809
$293$	6.79614	0.397035	0.198517	−	0.980097i	$-0.436387\pi$
$293$	6.79614	0.397035	0.198517	+	0.980097i	$0.436387\pi$
$294$	0	0
$295$	33.2626	1.93662
$296$	−14.4094	−0.837528
$297$	0	0
$298$	7.20688	0.417484
$299$	−9.84170	−0.569160
$300$	0	0
$301$	6.68415	0.385268
$302$	−34.7549	−1.99992
$303$	0	0
$304$	0.379268	0.0217525
$305$	−3.00893	−0.172291
$306$	0	0
$307$	−4.53796	−0.258995	−0.129497	−	0.991580i	$-0.541336\pi$
$307$	−4.53796	−0.258995	−0.129497	+	0.991580i	$0.541336\pi$
$308$	−5.43303	−0.309576
$309$	0	0
$310$	−9.21088	−0.523143
$311$	7.76550	0.440341	0.220171	−	0.975461i	$-0.429339\pi$
$311$	7.76550	0.440341	0.220171	+	0.975461i	$0.429339\pi$
$312$	0	0
$313$	−25.7688	−1.45654	−0.728268	−	0.685292i	$-0.759675\pi$
$313$	−25.7688	−1.45654	−0.728268	+	0.685292i	$0.759675\pi$
$314$	−55.7524	−3.14629
$315$	0	0
$316$	21.0743	1.18552
$317$	−0.0945206	−0.00530881	−0.00265440	−	0.999996i	$-0.500845\pi$
$317$	−0.0945206	−0.00530881	−0.00265440	+	0.999996i	$0.500845\pi$
$318$	0	0
$319$	4.08300	0.228604
$320$	−46.4091	−2.59435
$321$	0	0
$322$	9.73905	0.542736
$323$	−0.0701869	−0.00390531
$324$	0	0
$325$	−15.1115	−0.838235
$326$	7.83784	0.434098
$327$	0	0
$328$	−11.2735	−0.622476
$329$	0.858342	0.0473219
$330$	0	0
$331$	20.2893	1.11520	0.557600	−	0.830110i	$-0.311722\pi$
$331$	20.2893	1.11520	0.557600	+	0.830110i	$0.311722\pi$
$332$	−26.4037	−1.44909
$333$	0	0
$334$	−42.3348	−2.31645
$335$	10.5413	0.575931
$336$	0	0
$337$	−21.2726	−1.15879	−0.579395	−	0.815047i	$-0.696711\pi$
$337$	−21.2726	−1.15879	−0.579395	+	0.815047i	$0.696711\pi$
$338$	20.7085	1.12640
$339$	0	0
$340$	0.787088	0.0426858
$341$	2.29102	0.124066
$342$	0	0
$343$	11.3844	0.614700
$344$	20.3149	1.09531
$345$	0	0
$346$	−5.30352	−0.285119
$347$	1.75562	0.0942467	0.0471234	−	0.998889i	$-0.484995\pi$
$347$	1.75562	0.0942467	0.0471234	+	0.998889i	$0.484995\pi$
$348$	0	0
$349$	15.6718	0.838890	0.419445	−	0.907781i	$-0.362225\pi$
$349$	15.6718	0.838890	0.419445	+	0.907781i	$0.362225\pi$
$350$	14.9539	0.799319
$351$	0	0
$352$	12.2151	0.651067
$353$	−12.1480	−0.646574	−0.323287	−	0.946301i	$-0.604788\pi$
$353$	−12.1480	−0.646574	−0.323287	+	0.946301i	$0.604788\pi$
$354$	0	0
$355$	3.72737	0.197828
$356$	−32.5556	−1.72544
$357$	0	0
$358$	−24.8253	−1.31206
$359$	−1.07096	−0.0565230	−0.0282615	−	0.999601i	$-0.508997\pi$
$359$	−1.07096	−0.0565230	−0.0282615	+	0.999601i	$0.508997\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−27.8054	−1.46142
$363$	0	0
$364$	5.32126	0.278910
$365$	−0.391776	−0.0205065
$366$	0	0
$367$	10.0683	0.525560	0.262780	−	0.964856i	$-0.415361\pi$
$367$	10.0683	0.525560	0.262780	+	0.964856i	$0.415361\pi$
$368$	−1.89634	−0.0988535
$369$	0	0
$370$	−44.6289	−2.32014
$371$	−7.21827	−0.374754
$372$	0	0
$373$	17.8394	0.923691	0.461846	−	0.886960i	$-0.347187\pi$
$373$	17.8394	0.923691	0.461846	+	0.886960i	$0.347187\pi$
$374$	−0.320089	−0.0165514
$375$	0	0
$376$	2.60873	0.134535
$377$	−3.99900	−0.205959
$378$	0	0
$379$	1.17366	0.0602869	0.0301435	−	0.999546i	$-0.490404\pi$
$379$	1.17366	0.0602869	0.0301435	+	0.999546i	$0.490404\pi$
$380$	−11.2142	−0.575275
$381$	0	0
$382$	14.6736	0.750768
$383$	21.0327	1.07472	0.537361	−	0.843352i	$-0.319421\pi$
$383$	21.0327	1.07472	0.537361	+	0.843352i	$0.319421\pi$
$384$	0	0
$385$	−6.14188	−0.313019
$386$	14.5411	0.740125
$387$	0	0
$388$	−13.4930	−0.685005
$389$	6.16100	0.312375	0.156188	−	0.987727i	$-0.450080\pi$
$389$	6.16100	0.312375	0.156188	+	0.987727i	$0.450080\pi$
$390$	0	0
$391$	0.350935	0.0177475
$392$	16.3391	0.825250
$393$	0	0
$394$	36.6581	1.84681
$395$	23.8238	1.19871
$396$	0	0
$397$	6.21729	0.312037	0.156018	−	0.987754i	$-0.450134\pi$
$397$	6.21729	0.312037	0.156018	+	0.987754i	$0.450134\pi$
$398$	18.3893	0.921771
$399$	0	0
$400$	−2.91175	−0.145587
$401$	−15.8306	−0.790543	−0.395271	−	0.918564i	$-0.629349\pi$
$401$	−15.8306	−0.790543	−0.395271	+	0.918564i	$0.629349\pi$
$402$	0	0
$403$	−2.24389	−0.111776
$404$	−45.7519	−2.27624
$405$	0	0
$406$	3.95729	0.196397
$407$	11.1005	0.550234
$408$	0	0
$409$	−12.4064	−0.613459	−0.306730	−	0.951797i	$-0.599235\pi$
$409$	−12.4064	−0.613459	−0.306730	+	0.951797i	$0.599235\pi$
$410$	−34.9164	−1.72440
$411$	0	0
$412$	29.4146	1.44915
$413$	−8.01868	−0.394573
$414$	0	0
$415$	−29.8486	−1.46521
$416$	−11.9638	−0.586573
$417$	0	0
$418$	4.56052	0.223062
$419$	−13.6375	−0.666234	−0.333117	−	0.942885i	$-0.608100\pi$
$419$	−13.6375	−0.666234	−0.333117	+	0.942885i	$0.608100\pi$
$420$	0	0
$421$	3.45406	0.168340	0.0841702	−	0.996451i	$-0.473176\pi$
$421$	3.45406	0.168340	0.0841702	+	0.996451i	$0.473176\pi$
$422$	17.2962	0.841966
$423$	0	0
$424$	−21.9382	−1.06541
$425$	0.538845	0.0261378
$426$	0	0
$427$	0.725369	0.0351031
$428$	−37.4736	−1.81135
$429$	0	0
$430$	62.9195	3.03425
$431$	−9.08928	−0.437815	−0.218908	−	0.975746i	$-0.570249\pi$
$431$	−9.08928	−0.437815	−0.218908	+	0.975746i	$0.570249\pi$
$432$	0	0
$433$	16.9303	0.813617	0.406809	−	0.913513i	$-0.366642\pi$
$433$	16.9303	0.813617	0.406809	+	0.913513i	$0.366642\pi$
$434$	2.22049	0.106587
$435$	0	0
$436$	9.41302	0.450802
$437$	−5.00000	−0.239182
$438$	0	0
$439$	−14.6283	−0.698170	−0.349085	−	0.937091i	$-0.613508\pi$
$439$	−14.6283	−0.698170	−0.349085	+	0.937091i	$0.613508\pi$
$440$	−18.6668	−0.889905
$441$	0	0
$442$	0.313504	0.0149118
$443$	22.2264	1.05601	0.528004	−	0.849242i	$-0.322941\pi$
$443$	22.2264	1.05601	0.528004	+	0.849242i	$0.322941\pi$
$444$	0	0
$445$	−36.8031	−1.74463
$446$	−19.2696	−0.912444
$447$	0	0
$448$	11.1879	0.528580
$449$	−4.47539	−0.211207	−0.105603	−	0.994408i	$-0.533677\pi$
$449$	−4.47539	−0.211207	−0.105603	+	0.994408i	$0.533677\pi$
$450$	0	0
$451$	8.68477	0.408950
$452$	31.0501	1.46047
$453$	0	0
$454$	−41.9382	−1.96826
$455$	6.01552	0.282012
$456$	0	0
$457$	31.5512	1.47590	0.737952	−	0.674853i	$-0.235793\pi$
$457$	31.5512	1.47590	0.737952	+	0.674853i	$0.235793\pi$
$458$	8.80106	0.411246
$459$	0	0
$460$	56.0708	2.61432
$461$	−20.6204	−0.960386	−0.480193	−	0.877163i	$-0.659433\pi$
$461$	−20.6204	−0.960386	−0.480193	+	0.877163i	$0.659433\pi$
$462$	0	0
$463$	−39.7023	−1.84512	−0.922562	−	0.385850i	$-0.873908\pi$
$463$	−39.7023	−1.84512	−0.922562	+	0.385850i	$0.873908\pi$
$464$	−0.770544	−0.0357716
$465$	0	0
$466$	−38.6974	−1.79262
$467$	7.54017	0.348917	0.174459	−	0.984665i	$-0.444182\pi$
$467$	7.54017	0.348917	0.174459	+	0.984665i	$0.444182\pi$
$468$	0	0
$469$	−2.54121	−0.117342
$470$	8.07978	0.372692
$471$	0	0
$472$	−24.3709	−1.12176
$473$	−15.6500	−0.719587
$474$	0	0
$475$	−7.67728	−0.352258
$476$	−0.189745	−0.00869695
$477$	0	0
$478$	−53.3873	−2.44188
$479$	7.24294	0.330938	0.165469	−	0.986215i	$-0.447086\pi$
$479$	7.24294	0.330938	0.165469	+	0.986215i	$0.447086\pi$
$480$	0	0
$481$	−10.8722	−0.495729
$482$	49.9608	2.27565
$483$	0	0
$484$	−21.9248	−0.996583
$485$	−15.2535	−0.692625
$486$	0	0
$487$	7.34998	0.333059	0.166530	−	0.986036i	$-0.446744\pi$
$487$	7.34998	0.333059	0.166530	+	0.986036i	$0.446744\pi$
$488$	2.20459	0.0997971
$489$	0	0
$490$	50.6057	2.28613
$491$	−15.7146	−0.709189	−0.354594	−	0.935020i	$-0.615381\pi$
$491$	−15.7146	−0.709189	−0.354594	+	0.935020i	$0.615381\pi$
$492$	0	0
$493$	0.142596	0.00642220
$494$	−4.46669	−0.200966
$495$	0	0
$496$	−0.432362	−0.0194136
$497$	−0.898565	−0.0403061
$498$	0	0
$499$	−3.11598	−0.139491	−0.0697453	−	0.997565i	$-0.522219\pi$
$499$	−3.11598	−0.139491	−0.0697453	+	0.997565i	$0.522219\pi$
$500$	30.0235	1.34269
$501$	0	0
$502$	2.06089	0.0919822
$503$	2.40339	0.107162	0.0535810	−	0.998564i	$-0.482936\pi$
$503$	2.40339	0.107162	0.0535810	+	0.998564i	$0.482936\pi$
$504$	0	0
$505$	−51.7212	−2.30156
$506$	−22.8026	−1.01370
$507$	0	0
$508$	22.3179	0.990198
$509$	33.5019	1.48495	0.742473	−	0.669876i	$-0.233653\pi$
$509$	33.5019	1.48495	0.742473	+	0.669876i	$0.233653\pi$
$510$	0	0
$511$	0.0944463	0.00417806
$512$	−4.28405	−0.189330
$513$	0	0
$514$	43.4105	1.91476
$515$	33.2523	1.46527
$516$	0	0
$517$	−2.00968	−0.0883858
$518$	10.7588	0.472714
$519$	0	0
$520$	18.2828	0.801752
$521$	26.2833	1.15149	0.575747	−	0.817628i	$-0.304712\pi$
$521$	26.2833	1.15149	0.575747	+	0.817628i	$0.304712\pi$
$522$	0	0
$523$	−32.4592	−1.41934	−0.709672	−	0.704533i	$-0.751157\pi$
$523$	−32.4592	−1.41934	−0.709672	+	0.704533i	$0.751157\pi$
$524$	18.7596	0.819517
$525$	0	0
$526$	−40.7151	−1.77526
$527$	0.0800126	0.00348540
$528$	0	0
$529$	2.00000	0.0869565
$530$	−67.9473	−2.95144
$531$	0	0
$532$	2.70342	0.117208
$533$	−8.50610	−0.368440
$534$	0	0
$535$	−42.3628	−1.83150
$536$	−7.72340	−0.333600
$537$	0	0
$538$	9.88937	0.426361
$539$	−12.5872	−0.542167
$540$	0	0
$541$	−44.8410	−1.92786	−0.963932	−	0.266150i	$-0.914248\pi$
$541$	−44.8410	−1.92786	−0.963932	+	0.266150i	$0.914248\pi$
$542$	2.25290	0.0967702
$543$	0	0
$544$	0.426605	0.0182905
$545$	10.6411	0.455816
$546$	0	0
$547$	−38.8446	−1.66087	−0.830437	−	0.557112i	$-0.811909\pi$
$547$	−38.8446	−1.66087	−0.830437	+	0.557112i	$0.811909\pi$
$548$	43.9238	1.87633
$549$	0	0
$550$	−35.0124	−1.49293
$551$	−2.03166	−0.0865517
$552$	0	0
$553$	−5.74326	−0.244228
$554$	23.2276	0.986847
$555$	0	0
$556$	25.2116	1.06921
$557$	5.44557	0.230736	0.115368	−	0.993323i	$-0.463195\pi$
$557$	5.44557	0.230736	0.115368	+	0.993323i	$0.463195\pi$
$558$	0	0
$559$	15.3280	0.648306
$560$	1.15910	0.0489808
$561$	0	0
$562$	−41.9895	−1.77122
$563$	−12.6474	−0.533023	−0.266511	−	0.963832i	$-0.585871\pi$
$563$	−12.6474	−0.533023	−0.266511	+	0.963832i	$0.585871\pi$
$564$	0	0
$565$	35.1012	1.47672
$566$	−17.1155	−0.719420
$567$	0	0
$568$	−2.73098	−0.114589
$569$	22.2403	0.932360	0.466180	−	0.884690i	$-0.345630\pi$
$569$	22.2403	0.932360	0.466180	+	0.884690i	$0.345630\pi$
$570$	0	0
$571$	−37.3327	−1.56233	−0.781163	−	0.624327i	$-0.785373\pi$
$571$	−37.3327	−1.56233	−0.781163	+	0.624327i	$0.785373\pi$
$572$	−12.4590	−0.520935
$573$	0	0
$574$	8.41738	0.351335
$575$	38.3864	1.60082
$576$	0	0
$577$	28.4412	1.18402	0.592011	−	0.805930i	$-0.298334\pi$
$577$	28.4412	1.18402	0.592011	+	0.805930i	$0.298334\pi$
$578$	38.5664	1.60415
$579$	0	0
$580$	22.7834	0.946028
$581$	7.19567	0.298527
$582$	0	0
$583$	16.9005	0.699949
$584$	0.287047	0.0118781
$585$	0	0
$586$	−15.4223	−0.637089
$587$	33.7678	1.39375	0.696873	−	0.717194i	$-0.254574\pi$
$587$	33.7678	1.39375	0.696873	+	0.717194i	$0.254574\pi$
$588$	0	0
$589$	−1.13999	−0.0469726
$590$	−75.4818	−3.10753
$591$	0	0
$592$	−2.09490	−0.0860997
$593$	−44.1008	−1.81100	−0.905502	−	0.424341i	$-0.860506\pi$
$593$	−44.1008	−1.81100	−0.905502	+	0.424341i	$0.860506\pi$
$594$	0	0
$595$	−0.214501	−0.00879369
$596$	−10.0026	−0.409724
$597$	0	0
$598$	22.3335	0.913284
$599$	9.17373	0.374828	0.187414	−	0.982281i	$-0.439989\pi$
$599$	9.17373	0.374828	0.187414	+	0.982281i	$0.439989\pi$
$600$	0	0
$601$	−9.59099	−0.391224	−0.195612	−	0.980681i	$-0.562669\pi$
$601$	−9.59099	−0.391224	−0.195612	+	0.980681i	$0.562669\pi$
$602$	−15.1681	−0.618207
$603$	0	0
$604$	48.2374	1.96275
$605$	−24.7854	−1.00767
$606$	0	0
$607$	−20.3037	−0.824100	−0.412050	−	0.911161i	$-0.635187\pi$
$607$	−20.3037	−0.824100	−0.412050	+	0.911161i	$0.635187\pi$
$608$	−6.07812	−0.246500
$609$	0	0
$610$	6.82807	0.276461
$611$	1.96834	0.0796305
$612$	0	0
$613$	10.8405	0.437842	0.218921	−	0.975743i	$-0.429746\pi$
$613$	10.8405	0.437842	0.218921	+	0.975743i	$0.429746\pi$
$614$	10.2979	0.415587
$615$	0	0
$616$	4.50005	0.181312
$617$	44.6835	1.79889	0.899445	−	0.437034i	$-0.143971\pi$
$617$	44.6835	1.79889	0.899445	+	0.437034i	$0.143971\pi$
$618$	0	0
$619$	0.609194	0.0244856	0.0122428	−	0.999925i	$-0.496103\pi$
$619$	0.609194	0.0244856	0.0122428	+	0.999925i	$0.496103\pi$
$620$	12.7841	0.513420
$621$	0	0
$622$	−17.6220	−0.706579
$623$	8.87221	0.355457
$624$	0	0
$625$	−4.44572	−0.177829
$626$	58.4763	2.33718
$627$	0	0
$628$	77.3804	3.08781
$629$	0.387680	0.0154578
$630$	0	0
$631$	24.8724	0.990155	0.495077	−	0.868849i	$-0.335140\pi$
$631$	24.8724	0.990155	0.495077	+	0.868849i	$0.335140\pi$
$632$	−17.4553	−0.694334
$633$	0	0
$634$	0.214493	0.00851860
$635$	25.2298	1.00121
$636$	0	0
$637$	12.3282	0.488461
$638$	−9.26542	−0.366822
$639$	0	0
$640$	62.0322	2.45204
$641$	−22.7956	−0.900372	−0.450186	−	0.892935i	$-0.648642\pi$
$641$	−22.7956	−0.900372	−0.450186	+	0.892935i	$0.648642\pi$
$642$	0	0
$643$	32.7197	1.29034	0.645170	−	0.764039i	$-0.276787\pi$
$643$	32.7197	1.29034	0.645170	+	0.764039i	$0.276787\pi$
$644$	−13.5171	−0.532649
$645$	0	0
$646$	0.159273	0.00626652
$647$	9.04964	0.355778	0.177889	−	0.984051i	$-0.443073\pi$
$647$	9.04964	0.355778	0.177889	+	0.984051i	$0.443073\pi$
$648$	0	0
$649$	18.7746	0.736967
$650$	34.2921	1.34505
$651$	0	0
$652$	−10.8784	−0.426030
$653$	33.5510	1.31295	0.656476	−	0.754347i	$-0.272046\pi$
$653$	33.5510	1.31295	0.656476	+	0.754347i	$0.272046\pi$
$654$	0	0
$655$	21.2072	0.828633
$656$	−1.63899	−0.0639919
$657$	0	0
$658$	−1.94781	−0.0759335
$659$	−8.91074	−0.347113	−0.173557	−	0.984824i	$-0.555526\pi$
$659$	−8.91074	−0.347113	−0.173557	+	0.984824i	$0.555526\pi$
$660$	0	0
$661$	50.2164	1.95319	0.976596	−	0.215084i	$-0.0690025\pi$
$661$	50.2164	1.95319	0.976596	+	0.215084i	$0.0690025\pi$
$662$	−46.0419	−1.78947
$663$	0	0
$664$	21.8695	0.848703
$665$	3.05614	0.118512
$666$	0	0
$667$	10.1583	0.393331
$668$	58.7577	2.27340
$669$	0	0
$670$	−23.9210	−0.924149
$671$	−1.69835	−0.0655640
$672$	0	0
$673$	−2.74564	−0.105837	−0.0529183	−	0.998599i	$-0.516852\pi$
$673$	−2.74564	−0.105837	−0.0529183	+	0.998599i	$0.516852\pi$
$674$	48.2732	1.85941
$675$	0	0
$676$	−28.7420	−1.10546
$677$	6.64803	0.255505	0.127752	−	0.991806i	$-0.459224\pi$
$677$	6.64803	0.255505	0.127752	+	0.991806i	$0.459224\pi$
$678$	0	0
$679$	3.67719	0.141118
$680$	−0.651926	−0.0250002
$681$	0	0
$682$	−5.19895	−0.199078
$683$	−37.5156	−1.43550	−0.717748	−	0.696303i	$-0.754827\pi$
$683$	−37.5156	−1.43550	−0.717748	+	0.696303i	$0.754827\pi$
$684$	0	0
$685$	49.6545	1.89720
$686$	−25.8343	−0.986358
$687$	0	0
$688$	2.95347	0.112600
$689$	−16.5528	−0.630613
$690$	0	0
$691$	−6.26564	−0.238356	−0.119178	−	0.992873i	$-0.538026\pi$
$691$	−6.26564	−0.238356	−0.119178	+	0.992873i	$0.538026\pi$
$692$	7.36092	0.279820
$693$	0	0
$694$	−3.98398	−0.151230
$695$	28.5010	1.08110
$696$	0	0
$697$	0.303310	0.0114887
$698$	−35.5635	−1.34610
$699$	0	0
$700$	−20.7550	−0.784464
$701$	38.4046	1.45052	0.725261	−	0.688474i	$-0.241719\pi$
$701$	38.4046	1.45052	0.725261	+	0.688474i	$0.241719\pi$
$702$	0	0
$703$	−5.52353	−0.208324
$704$	−26.1949	−0.987259
$705$	0	0
$706$	27.5671	1.03750
$707$	12.4685	0.468928
$708$	0	0
$709$	6.16430	0.231505	0.115753	−	0.993278i	$-0.463072\pi$
$709$	6.16430	0.231505	0.115753	+	0.993278i	$0.463072\pi$
$710$	−8.45840	−0.317438
$711$	0	0
$712$	26.9650	1.01056
$713$	5.69996	0.213465
$714$	0	0
$715$	−14.0845	−0.526730
$716$	34.4557	1.28767
$717$	0	0
$718$	2.43029	0.0906977
$719$	−26.8894	−1.00281	−0.501403	−	0.865214i	$-0.667182\pi$
$719$	−26.8894	−1.00281	−0.501403	+	0.865214i	$0.667182\pi$
$720$	0	0
$721$	−8.01621	−0.298539
$722$	−2.26927	−0.0844535
$723$	0	0
$724$	38.5919	1.43426
$725$	15.5976	0.579282
$726$	0	0
$727$	14.1233	0.523802	0.261901	−	0.965095i	$-0.415651\pi$
$727$	14.1233	0.523802	0.261901	+	0.965095i	$0.415651\pi$
$728$	−4.40747	−0.163352
$729$	0	0
$730$	0.889045	0.0329050
$731$	−0.546566	−0.0202155
$732$	0	0
$733$	−4.54598	−0.167909	−0.0839547	−	0.996470i	$-0.526755\pi$
$733$	−4.54598	−0.167909	−0.0839547	+	0.996470i	$0.526755\pi$
$734$	−22.8477	−0.843322
$735$	0	0
$736$	30.3906	1.12021
$737$	5.94987	0.219166
$738$	0	0
$739$	−21.3470	−0.785260	−0.392630	−	0.919696i	$-0.628435\pi$
$739$	−21.3470	−0.785260	−0.392630	+	0.919696i	$0.628435\pi$
$740$	61.9418	2.27702
$741$	0	0
$742$	16.3802	0.601336
$743$	−26.3700	−0.967421	−0.483711	−	0.875228i	$-0.660711\pi$
$743$	−26.3700	−0.967421	−0.483711	+	0.875228i	$0.660711\pi$
$744$	0	0
$745$	−11.3077	−0.414282
$746$	−40.4825	−1.48217
$747$	0	0
$748$	0.444261	0.0162438
$749$	10.2125	0.373156
$750$	0	0
$751$	−3.68059	−0.134307	−0.0671534	−	0.997743i	$-0.521392\pi$
$751$	−3.68059	−0.134307	−0.0671534	+	0.997743i	$0.521392\pi$
$752$	0.379268	0.0138305
$753$	0	0
$754$	9.07481	0.330485
$755$	54.5309	1.98458
$756$	0	0
$757$	30.7404	1.11728	0.558640	−	0.829410i	$-0.311323\pi$
$757$	30.7404	1.11728	0.558640	+	0.829410i	$0.311323\pi$
$758$	−2.66336	−0.0967374
$759$	0	0
$760$	9.28842	0.336926
$761$	18.9350	0.686394	0.343197	−	0.939263i	$-0.388490\pi$
$761$	18.9350	0.686394	0.343197	+	0.939263i	$0.388490\pi$
$762$	0	0
$763$	−2.56528	−0.0928695
$764$	−20.3660	−0.736814
$765$	0	0
$766$	−47.7290	−1.72452
$767$	−18.3883	−0.663965
$768$	0	0
$769$	−2.46397	−0.0888530	−0.0444265	−	0.999013i	$-0.514146\pi$
$769$	−2.46397	−0.0888530	−0.0444265	+	0.999013i	$0.514146\pi$
$770$	13.9376	0.502276
$771$	0	0
$772$	−20.1821	−0.726369
$773$	37.3168	1.34219	0.671096	−	0.741371i	$-0.265824\pi$
$773$	37.3168	1.34219	0.671096	+	0.741371i	$0.265824\pi$
$774$	0	0
$775$	8.75204	0.314383
$776$	11.1760	0.401194
$777$	0	0
$778$	−13.9810	−0.501243
$779$	−4.32146	−0.154832
$780$	0	0
$781$	2.10386	0.0752820
$782$	−0.796366	−0.0284780
$783$	0	0
$784$	2.37545	0.0848375
$785$	87.4763	3.12216
$786$	0	0
$787$	20.8520	0.743293	0.371647	−	0.928374i	$-0.378793\pi$
$787$	20.8520	0.743293	0.371647	+	0.928374i	$0.378793\pi$
$788$	−50.8789	−1.81249
$789$	0	0
$790$	−54.0627	−1.92346
$791$	−8.46193	−0.300872
$792$	0	0
$793$	1.66341	0.0590694
$794$	−14.1087	−0.500700
$795$	0	0
$796$	−25.5230	−0.904640
$797$	−3.05491	−0.108211	−0.0541053	−	0.998535i	$-0.517231\pi$
$797$	−3.05491	−0.108211	−0.0541053	+	0.998535i	$0.517231\pi$
$798$	0	0
$799$	−0.0701869	−0.00248304
$800$	46.6634	1.64980
$801$	0	0
$802$	35.9239	1.26852
$803$	−0.221132	−0.00780359
$804$	0	0
$805$	−15.2807	−0.538574
$806$	5.09200	0.179358
$807$	0	0
$808$	37.8952	1.33315
$809$	−3.00256	−0.105564	−0.0527822	−	0.998606i	$-0.516809\pi$
$809$	−3.00256	−0.105564	−0.0527822	+	0.998606i	$0.516809\pi$
$810$	0	0
$811$	32.1171	1.12778	0.563891	−	0.825849i	$-0.309304\pi$
$811$	32.1171	1.12778	0.563891	+	0.825849i	$0.309304\pi$
$812$	−5.49244	−0.192747
$813$	0	0
$814$	−25.1901	−0.882914
$815$	−12.2977	−0.430769
$816$	0	0
$817$	7.78728	0.272443
$818$	28.1536	0.984367
$819$	0	0
$820$	48.4616	1.69235
$821$	−24.4194	−0.852243	−0.426122	−	0.904666i	$-0.640120\pi$
$821$	−24.4194	−0.852243	−0.426122	+	0.904666i	$0.640120\pi$
$822$	0	0
$823$	6.69671	0.233433	0.116716	−	0.993165i	$-0.462763\pi$
$823$	6.69671	0.233433	0.116716	+	0.993165i	$0.462763\pi$
$824$	−24.3634	−0.848738
$825$	0	0
$826$	18.1966	0.633139
$827$	−11.3210	−0.393670	−0.196835	−	0.980437i	$-0.563066\pi$
$827$	−11.3210	−0.393670	−0.196835	+	0.980437i	$0.563066\pi$
$828$	0	0
$829$	−8.77277	−0.304691	−0.152345	−	0.988327i	$-0.548683\pi$
$829$	−8.77277	−0.304691	−0.152345	+	0.988327i	$0.548683\pi$
$830$	67.7345	2.35110
$831$	0	0
$832$	25.6560	0.889463
$833$	−0.439598	−0.0152312
$834$	0	0
$835$	66.4238	2.29869
$836$	−6.32968	−0.218917
$837$	0	0
$838$	30.9471	1.06905
$839$	4.80157	0.165768	0.0828842	−	0.996559i	$-0.473587\pi$
$839$	4.80157	0.165768	0.0828842	+	0.996559i	$0.473587\pi$
$840$	0	0
$841$	−24.8724	−0.857667
$842$	−7.83819	−0.270122
$843$	0	0
$844$	−24.0059	−0.826318
$845$	−32.4920	−1.11776
$846$	0	0
$847$	5.97506	0.205306
$848$	−3.18947	−0.109527
$849$	0	0
$850$	−1.22279	−0.0419412
$851$	27.6176	0.946720
$852$	0	0
$853$	9.89266	0.338718	0.169359	−	0.985554i	$-0.445830\pi$
$853$	9.89266	0.338718	0.169359	+	0.985554i	$0.445830\pi$
$854$	−1.64606	−0.0563270
$855$	0	0
$856$	31.0385	1.06087
$857$	24.0020	0.819892	0.409946	−	0.912110i	$-0.365548\pi$
$857$	24.0020	0.819892	0.409946	+	0.912110i	$0.365548\pi$
$858$	0	0
$859$	31.4087	1.07165	0.535825	−	0.844329i	$-0.320001\pi$
$859$	31.4087	1.07165	0.535825	+	0.844329i	$0.320001\pi$
$860$	−87.3279	−2.97786
$861$	0	0
$862$	20.6260	0.702525
$863$	33.4135	1.13741	0.568705	−	0.822542i	$-0.307445\pi$
$863$	33.4135	1.13741	0.568705	+	0.822542i	$0.307445\pi$
$864$	0	0
$865$	8.32130	0.282933
$866$	−38.4194	−1.30554
$867$	0	0
$868$	−3.08188	−0.104606
$869$	13.4470	0.456159
$870$	0	0
$871$	−5.82747	−0.197456
$872$	−7.79657	−0.264025
$873$	0	0
$874$	11.3464	0.383796
$875$	−8.18216	−0.276608
$876$	0	0
$877$	−24.2149	−0.817680	−0.408840	−	0.912606i	$-0.634067\pi$
$877$	−24.2149	−0.817680	−0.408840	+	0.912606i	$0.634067\pi$
$878$	33.1955	1.12030
$879$	0	0
$880$	−2.71386	−0.0914841
$881$	−40.6329	−1.36896	−0.684478	−	0.729033i	$-0.739970\pi$
$881$	−40.6329	−1.36896	−0.684478	+	0.729033i	$0.739970\pi$
$882$	0	0
$883$	7.63383	0.256899	0.128449	−	0.991716i	$-0.459000\pi$
$883$	7.63383	0.256899	0.128449	+	0.991716i	$0.459000\pi$
$884$	−0.435121	−0.0146347
$885$	0	0
$886$	−50.4377	−1.69449
$887$	−33.6851	−1.13103	−0.565517	−	0.824736i	$-0.691324\pi$
$887$	−33.6851	−1.13103	−0.565517	+	0.824736i	$0.691324\pi$
$888$	0	0
$889$	−6.08219	−0.203990
$890$	83.5162	2.79947
$891$	0	0
$892$	26.7449	0.895486
$893$	1.00000	0.0334637
$894$	0	0
$895$	38.9512	1.30199
$896$	−14.9542	−0.499586
$897$	0	0
$898$	10.1559	0.338906
$899$	2.31608	0.0772455
$900$	0	0
$901$	0.590241	0.0196638
$902$	−19.7081	−0.656208
$903$	0	0
$904$	−25.7181	−0.855370
$905$	43.6270	1.45021
$906$	0	0
$907$	21.6878	0.720131	0.360065	−	0.932927i	$-0.382754\pi$
$907$	21.6878	0.720131	0.360065	+	0.932927i	$0.382754\pi$
$908$	58.2073	1.93168
$909$	0	0
$910$	−13.6508	−0.452521
$911$	47.6262	1.57793	0.788964	−	0.614439i	$-0.210618\pi$
$911$	47.6262	1.57793	0.788964	+	0.614439i	$0.210618\pi$
$912$	0	0
$913$	−16.8476	−0.557575
$914$	−71.5983	−2.36826
$915$	0	0
$916$	−12.2153	−0.403603
$917$	−5.11246	−0.168828
$918$	0	0
$919$	23.9686	0.790652	0.395326	−	0.918541i	$-0.370632\pi$
$919$	23.9686	0.790652	0.395326	+	0.918541i	$0.370632\pi$
$920$	−46.4421	−1.53115
$921$	0	0
$922$	46.7932	1.54105
$923$	−2.06058	−0.0678247
$924$	0	0
$925$	42.4057	1.39429
$926$	90.0953	2.96072
$927$	0	0
$928$	12.3487	0.405365
$929$	16.4686	0.540318	0.270159	−	0.962816i	$-0.412924\pi$
$929$	16.4686	0.540318	0.270159	+	0.962816i	$0.412924\pi$
$930$	0	0
$931$	6.26325	0.205270
$932$	53.7093	1.75931
$933$	0	0
$934$	−17.1107	−0.559878
$935$	0.502224	0.0164245
$936$	0	0
$937$	59.4601	1.94248	0.971238	−	0.238109i	$-0.0765276\pi$
$937$	59.4601	1.94248	0.971238	+	0.238109i	$0.0765276\pi$
$938$	5.76669	0.188289
$939$	0	0
$940$	−11.2142	−0.365766
$941$	−12.4691	−0.406480	−0.203240	−	0.979129i	$-0.565147\pi$
$941$	−12.4691	−0.406480	−0.203240	+	0.979129i	$0.565147\pi$
$942$	0	0
$943$	21.6073	0.703631
$944$	−3.54314	−0.115319
$945$	0	0
$946$	35.5140	1.15466
$947$	35.9027	1.16668	0.583340	−	0.812228i	$-0.301745\pi$
$947$	35.9027	1.16668	0.583340	+	0.812228i	$0.301745\pi$
$948$	0	0
$949$	0.216583	0.00703058
$950$	17.4218	0.565239
$951$	0	0
$952$	0.157161	0.00509362
$953$	0.387077	0.0125386	0.00626932	−	0.999980i	$-0.498004\pi$
$953$	0.387077	0.0125386	0.00626932	+	0.999980i	$0.498004\pi$
$954$	0	0
$955$	−23.0231	−0.745011
$956$	74.0979	2.39650
$957$	0	0
$958$	−16.4362	−0.531029
$959$	−11.9703	−0.386542
$960$	0	0
$961$	−29.7004	−0.958078
$962$	24.6719	0.795454
$963$	0	0
$964$	−69.3421	−2.23336
$965$	−22.8153	−0.734449
$966$	0	0
$967$	32.1625	1.03428	0.517139	−	0.855902i	$-0.326997\pi$
$967$	32.1625	1.03428	0.517139	+	0.855902i	$0.326997\pi$
$968$	18.1598	0.583678
$969$	0	0
$970$	34.6143	1.11140
$971$	8.79772	0.282332	0.141166	−	0.989986i	$-0.454915\pi$
$971$	8.79772	0.282332	0.141166	+	0.989986i	$0.454915\pi$
$972$	0	0
$973$	−6.87080	−0.220268
$974$	−16.6791	−0.534432
$975$	0	0
$976$	0.320513	0.0102594
$977$	31.2851	1.00090	0.500450	−	0.865766i	$-0.333168\pi$
$977$	31.2851	1.00090	0.500450	+	0.865766i	$0.333168\pi$
$978$	0	0
$979$	−20.7730	−0.663908
$980$	−70.2371	−2.24364
$981$	0	0
$982$	35.6606	1.13798
$983$	24.4098	0.778551	0.389276	−	0.921121i	$-0.372725\pi$
$983$	24.4098	0.778551	0.389276	+	0.921121i	$0.372725\pi$
$984$	0	0
$985$	−57.5171	−1.83265
$986$	−0.323589	−0.0103052
$987$	0	0
$988$	6.19946	0.197231
$989$	−38.9364	−1.23811
$990$	0	0
$991$	−47.6810	−1.51464	−0.757318	−	0.653046i	$-0.773491\pi$
$991$	−47.6810	−1.51464	−0.757318	+	0.653046i	$0.773491\pi$
$992$	6.92901	0.219996
$993$	0	0
$994$	2.03909	0.0646759
$995$	−28.8530	−0.914703
$996$	0	0
$997$	27.8996	0.883591	0.441795	−	0.897116i	$-0.354342\pi$
$997$	27.8996	0.883591	0.441795	+	0.897116i	$0.354342\pi$
$998$	7.07101	0.223829
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.k.1.1		7
3.2	odd	2		2679.2.a.l.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.l.1.7	✓	7	3.2	odd	2
8037.2.a.k.1.1		7	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)

\(f(q)\)	\(=\)	\(q-2.26927 q^{2} +3.14959 q^{4} +3.56052 q^{5} -0.858342 q^{7} -2.60873 q^{8} +O(q^{10})\)	\(q-2.26927 q^{2} +3.14959 q^{4} +3.56052 q^{5} -0.858342 q^{7} -2.60873 q^{8} -8.07978 q^{10} +2.00968 q^{11} -1.96834 q^{13} +1.94781 q^{14} -0.379268 q^{16} +0.0701869 q^{17} -1.00000 q^{19} +11.2142 q^{20} -4.56052 q^{22} +5.00000 q^{23} +7.67728 q^{25} +4.46669 q^{26} -2.70342 q^{28} +2.03166 q^{29} +1.13999 q^{31} +6.07812 q^{32} -0.159273 q^{34} -3.05614 q^{35} +5.52353 q^{37} +2.26927 q^{38} -9.28842 q^{40} +4.32146 q^{41} -7.78728 q^{43} +6.32968 q^{44} -11.3464 q^{46} -1.00000 q^{47} -6.26325 q^{49} -17.4218 q^{50} -6.19946 q^{52} +8.40955 q^{53} +7.15552 q^{55} +2.23918 q^{56} -4.61039 q^{58} +9.34206 q^{59} -0.845082 q^{61} -2.58695 q^{62} -13.0344 q^{64} -7.00831 q^{65} +2.96060 q^{67} +0.221060 q^{68} +6.93521 q^{70} +1.04686 q^{71} -0.110033 q^{73} -12.5344 q^{74} -3.14959 q^{76} -1.72500 q^{77} +6.69111 q^{79} -1.35039 q^{80} -9.80656 q^{82} -8.38322 q^{83} +0.249902 q^{85} +17.6715 q^{86} -5.24272 q^{88} -10.3364 q^{89} +1.68951 q^{91} +15.7479 q^{92} +2.26927 q^{94} -3.56052 q^{95} -4.28407 q^{97} +14.2130 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 8 q^{4} + 6 q^{5} + 7 q^{7} + 12 q^{8}+O(q^{10}) \) 7 * q + 2 * q^2 + 8 * q^4 + 6 * q^5 + 7 * q^7 + 12 * q^8	\( 7 q + 2 q^{2} + 8 q^{4} + 6 q^{5} + 7 q^{7} + 12 q^{8} - 2 q^{10} + 9 q^{11} - 2 q^{13} + 6 q^{14} + 14 q^{16} + 6 q^{17} - 7 q^{19} + 25 q^{20} - 13 q^{22} + 35 q^{23} + 3 q^{25} + 23 q^{26} - 9 q^{28} + 26 q^{29} - 11 q^{31} + 21 q^{32} + 18 q^{34} + 17 q^{35} - 3 q^{37} - 2 q^{38} + 14 q^{40} + 20 q^{41} - 5 q^{43} - 18 q^{44} + 10 q^{46} - 7 q^{47} - 10 q^{49} - 17 q^{50} + 15 q^{52} - 3 q^{53} - 3 q^{55} + 3 q^{56} + 31 q^{58} + 15 q^{59} - 9 q^{61} + 4 q^{62} + 4 q^{64} + 13 q^{65} - 4 q^{67} - 5 q^{68} + 18 q^{70} + 4 q^{71} - 8 q^{73} - 28 q^{74} - 8 q^{76} + 11 q^{77} - 11 q^{79} + 12 q^{80} - 19 q^{82} + 4 q^{83} - 6 q^{85} + 36 q^{86} - 7 q^{88} + 23 q^{89} + 19 q^{91} + 40 q^{92} - 2 q^{94} - 6 q^{95} - 7 q^{97}+O(q^{100}) \) 7 * q + 2 * q^2 + 8 * q^4 + 6 * q^5 + 7 * q^7 + 12 * q^8 - 2 * q^10 + 9 * q^11 - 2 * q^13 + 6 * q^14 + 14 * q^16 + 6 * q^17 - 7 * q^19 + 25 * q^20 - 13 * q^22 + 35 * q^23 + 3 * q^25 + 23 * q^26 - 9 * q^28 + 26 * q^29 - 11 * q^31 + 21 * q^32 + 18 * q^34 + 17 * q^35 - 3 * q^37 - 2 * q^38 + 14 * q^40 + 20 * q^41 - 5 * q^43 - 18 * q^44 + 10 * q^46 - 7 * q^47 - 10 * q^49 - 17 * q^50 + 15 * q^52 - 3 * q^53 - 3 * q^55 + 3 * q^56 + 31 * q^58 + 15 * q^59 - 9 * q^61 + 4 * q^62 + 4 * q^64 + 13 * q^65 - 4 * q^67 - 5 * q^68 + 18 * q^70 + 4 * q^71 - 8 * q^73 - 28 * q^74 - 8 * q^76 + 11 * q^77 - 11 * q^79 + 12 * q^80 - 19 * q^82 + 4 * q^83 - 6 * q^85 + 36 * q^86 - 7 * q^88 + 23 * q^89 + 19 * q^91 + 40 * q^92 - 2 * q^94 - 6 * q^95 - 7 * q^97

Introduction

L-functions

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Varieties

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Database

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