LMFDB - Embedded newform 8037.2.a.o.1.8

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			1.8
Root			$0.230018$ 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)

$f(q)$	$=$	$q-0.230018 q^{2} -1.94709 q^{4} +2.49148 q^{5} -0.810750 q^{7} +0.907901 q^{8} +O(q^{10})$	\(q-0.230018 q^{2} -1.94709 q^{4} +2.49148 q^{5} -0.810750 q^{7} +0.907901 q^{8} -0.573085 q^{10} +0.0355286 q^{11} -5.55359 q^{13} +0.186487 q^{14} +3.68535 q^{16} -4.02460 q^{17} -1.00000 q^{19} -4.85114 q^{20} -0.00817220 q^{22} +2.50005 q^{23} +1.20747 q^{25} +1.27742 q^{26} +1.57861 q^{28} -4.83006 q^{29} -2.28714 q^{31} -2.66350 q^{32} +0.925730 q^{34} -2.01997 q^{35} +7.13017 q^{37} +0.230018 q^{38} +2.26202 q^{40} -3.34464 q^{41} -2.39879 q^{43} -0.0691774 q^{44} -0.575056 q^{46} -1.00000 q^{47} -6.34268 q^{49} -0.277740 q^{50} +10.8133 q^{52} +7.50620 q^{53} +0.0885187 q^{55} -0.736081 q^{56} +1.11100 q^{58} -0.765058 q^{59} +9.17605 q^{61} +0.526083 q^{62} -6.75805 q^{64} -13.8366 q^{65} +7.91045 q^{67} +7.83627 q^{68} +0.464629 q^{70} +2.89523 q^{71} +13.6350 q^{73} -1.64007 q^{74} +1.94709 q^{76} -0.0288048 q^{77} -3.12962 q^{79} +9.18198 q^{80} +0.769326 q^{82} -9.75309 q^{83} -10.0272 q^{85} +0.551765 q^{86} +0.0322564 q^{88} +13.0683 q^{89} +4.50257 q^{91} -4.86783 q^{92} +0.230018 q^{94} -2.49148 q^{95} -0.389666 q^{97} +1.45893 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.230018	−0.162647	−0.0813236	−	0.996688i	$-0.525915\pi$
$2$	−0.230018	−0.162647	−0.0813236	+	0.996688i	$0.525915\pi$
$3$	0	0
$3$	0	0
$4$	−1.94709	−0.973546
$5$	2.49148	1.11422	0.557112	−	0.830437i	$-0.311909\pi$
$5$	2.49148	1.11422	0.557112	+	0.830437i	$0.311909\pi$
$6$	0	0
$7$	−0.810750	−0.306435	−0.153217	−	0.988193i	$-0.548963\pi$
$7$	−0.810750	−0.306435	−0.153217	+	0.988193i	$0.548963\pi$
$8$	0.907901	0.320992
$9$	0	0
$10$	−0.573085	−0.181225
$11$	0.0355286	0.0107123	0.00535613	−	0.999986i	$-0.498295\pi$
$11$	0.0355286	0.0107123	0.00535613	+	0.999986i	$0.498295\pi$
$12$	0	0
$13$	−5.55359	−1.54029	−0.770144	−	0.637870i	$-0.779816\pi$
$13$	−5.55359	−1.54029	−0.770144	+	0.637870i	$0.779816\pi$
$14$	0.186487	0.0498408
$15$	0	0
$16$	3.68535	0.921338
$17$	−4.02460	−0.976109	−0.488055	−	0.872813i	$-0.662293\pi$
$17$	−4.02460	−0.976109	−0.488055	+	0.872813i	$0.662293\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−4.85114	−1.08475
$21$	0	0
$22$	−0.00817220	−0.00174232
$23$	2.50005	0.521297	0.260648	−	0.965434i	$-0.416064\pi$
$23$	2.50005	0.521297	0.260648	+	0.965434i	$0.416064\pi$
$24$	0	0
$25$	1.20747	0.241495
$26$	1.27742	0.250523
$27$	0	0
$28$	1.57861	0.298328
$29$	−4.83006	−0.896920	−0.448460	−	0.893803i	$-0.648027\pi$
$29$	−4.83006	−0.896920	−0.448460	+	0.893803i	$0.648027\pi$
$30$	0	0
$31$	−2.28714	−0.410783	−0.205391	−	0.978680i	$-0.565847\pi$
$31$	−2.28714	−0.410783	−0.205391	+	0.978680i	$0.565847\pi$
$32$	−2.66350	−0.470845
$33$	0	0
$34$	0.925730	0.158761
$35$	−2.01997	−0.341437
$36$	0	0
$37$	7.13017	1.17219	0.586096	−	0.810241i	$-0.300664\pi$
$37$	7.13017	1.17219	0.586096	+	0.810241i	$0.300664\pi$
$38$	0.230018	0.0373138
$39$	0	0
$40$	2.26202	0.357656
$41$	−3.34464	−0.522344	−0.261172	−	0.965292i	$-0.584109\pi$
$41$	−3.34464	−0.522344	−0.261172	+	0.965292i	$0.584109\pi$
$42$	0	0
$43$	−2.39879	−0.365813	−0.182906	−	0.983130i	$-0.558550\pi$
$43$	−2.39879	−0.365813	−0.182906	+	0.983130i	$0.558550\pi$
$44$	−0.0691774	−0.0104289
$45$	0	0
$46$	−0.575056	−0.0847874
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−6.34268	−0.906098
$50$	−0.277740	−0.0392784
$51$	0	0
$52$	10.8133	1.49954
$53$	7.50620	1.03106	0.515528	−	0.856873i	$-0.327596\pi$
$53$	7.50620	1.03106	0.515528	+	0.856873i	$0.327596\pi$
$54$	0	0
$55$	0.0885187	0.0119359
$56$	−0.736081	−0.0983630
$57$	0	0
$58$	1.11100	0.145881
$59$	−0.765058	−0.0996021	−0.0498011	−	0.998759i	$-0.515859\pi$
$59$	−0.765058	−0.0996021	−0.0498011	+	0.998759i	$0.515859\pi$
$60$	0	0
$61$	9.17605	1.17487	0.587436	−	0.809270i	$-0.300137\pi$
$61$	9.17605	1.17487	0.587436	+	0.809270i	$0.300137\pi$
$62$	0.526083	0.0668126
$63$	0	0
$64$	−6.75805	−0.844756
$65$	−13.8366	−1.71622
$66$	0	0
$67$	7.91045	0.966416	0.483208	−	0.875506i	$-0.339472\pi$
$67$	7.91045	0.966416	0.483208	+	0.875506i	$0.339472\pi$
$68$	7.83627	0.950287
$69$	0	0
$70$	0.464629	0.0555338
$71$	2.89523	0.343600	0.171800	−	0.985132i	$-0.445042\pi$
$71$	2.89523	0.343600	0.171800	+	0.985132i	$0.445042\pi$
$72$	0	0
$73$	13.6350	1.59586	0.797930	−	0.602750i	$-0.205928\pi$
$73$	13.6350	1.59586	0.797930	+	0.602750i	$0.205928\pi$
$74$	−1.64007	−0.190654
$75$	0	0
$76$	1.94709	0.223347
$77$	−0.0288048	−0.00328261
$78$	0	0
$79$	−3.12962	−0.352110	−0.176055	−	0.984380i	$-0.556334\pi$
$79$	−3.12962	−0.352110	−0.176055	+	0.984380i	$0.556334\pi$
$80$	9.18198	1.02658
$81$	0	0
$82$	0.769326	0.0849578
$83$	−9.75309	−1.07054	−0.535270	−	0.844681i	$-0.679790\pi$
$83$	−9.75309	−1.07054	−0.535270	+	0.844681i	$0.679790\pi$
$84$	0	0
$85$	−10.0272	−1.08760
$86$	0.551765	0.0594984
$87$	0	0
$88$	0.0322564	0.00343855
$89$	13.0683	1.38524	0.692618	−	0.721304i	$-0.256457\pi$
$89$	13.0683	1.38524	0.692618	+	0.721304i	$0.256457\pi$
$90$	0	0
$91$	4.50257	0.471998
$92$	−4.86783	−0.507506
$93$	0	0
$94$	0.230018	0.0237245
$95$	−2.49148	−0.255620
$96$	0	0
$97$	−0.389666	−0.0395646	−0.0197823	−	0.999804i	$-0.506297\pi$
$97$	−0.389666	−0.0395646	−0.0197823	+	0.999804i	$0.506297\pi$
$98$	1.45893	0.147374
$99$	0	0
$100$	−2.35106	−0.235106
$101$	6.06968	0.603955	0.301978	−	0.953315i	$-0.402353\pi$
$101$	6.06968	0.603955	0.301978	+	0.953315i	$0.402353\pi$
$102$	0	0
$103$	−3.61182	−0.355884	−0.177942	−	0.984041i	$-0.556944\pi$
$103$	−3.61182	−0.355884	−0.177942	+	0.984041i	$0.556944\pi$
$104$	−5.04211	−0.494419
$105$	0	0
$106$	−1.72656	−0.167698
$107$	5.41031	0.523034	0.261517	−	0.965199i	$-0.415777\pi$
$107$	5.41031	0.523034	0.261517	+	0.965199i	$0.415777\pi$
$108$	0	0
$109$	6.45020	0.617817	0.308909	−	0.951092i	$-0.400036\pi$
$109$	6.45020	0.617817	0.308909	+	0.951092i	$0.400036\pi$
$110$	−0.0203609	−0.00194133
$111$	0	0
$112$	−2.98790	−0.282330
$113$	7.87210	0.740544	0.370272	−	0.928923i	$-0.379264\pi$
$113$	7.87210	0.740544	0.370272	+	0.928923i	$0.379264\pi$
$114$	0	0
$115$	6.22883	0.580841
$116$	9.40457	0.873193
$117$	0	0
$118$	0.175977	0.0162000
$119$	3.26295	0.299114
$120$	0	0
$121$	−10.9987	−0.999885
$122$	−2.11065	−0.191090
$123$	0	0
$124$	4.45327	0.399916
$125$	−9.44901	−0.845145
$126$	0	0
$127$	−12.2938	−1.09090	−0.545448	−	0.838144i	$-0.683641\pi$
$127$	−12.2938	−1.09090	−0.545448	+	0.838144i	$0.683641\pi$
$128$	6.88147	0.608242
$129$	0	0
$130$	3.18267	0.279139
$131$	8.08095	0.706036	0.353018	−	0.935617i	$-0.385155\pi$
$131$	8.08095	0.706036	0.353018	+	0.935617i	$0.385155\pi$
$132$	0	0
$133$	0.810750	0.0703010
$134$	−1.81955	−0.157185
$135$	0	0
$136$	−3.65394	−0.313323
$137$	7.56013	0.645906	0.322953	−	0.946415i	$-0.395324\pi$
$137$	7.56013	0.645906	0.322953	+	0.946415i	$0.395324\pi$
$138$	0	0
$139$	20.4854	1.73755	0.868775	−	0.495207i	$-0.164908\pi$
$139$	20.4854	1.73755	0.868775	+	0.495207i	$0.164908\pi$
$140$	3.93306	0.332405
$141$	0	0
$142$	−0.665954	−0.0558856
$143$	−0.197311	−0.0165000
$144$	0	0
$145$	−12.0340	−0.999370
$146$	−3.13630	−0.259562
$147$	0	0
$148$	−13.8831	−1.14118
$149$	−4.63106	−0.379391	−0.189695	−	0.981843i	$-0.560750\pi$
$149$	−4.63106	−0.379391	−0.189695	+	0.981843i	$0.560750\pi$
$150$	0	0
$151$	15.1229	1.23069	0.615343	−	0.788260i	$-0.289018\pi$
$151$	15.1229	1.23069	0.615343	+	0.788260i	$0.289018\pi$
$152$	−0.907901	−0.0736405
$153$	0	0
$154$	0.00662562	0.000533907 0
$155$	−5.69837	−0.457704
$156$	0	0
$157$	1.20753	0.0963710	0.0481855	−	0.998838i	$-0.484656\pi$
$157$	1.20753	0.0963710	0.0481855	+	0.998838i	$0.484656\pi$
$158$	0.719869	0.0572697
$159$	0	0
$160$	−6.63605	−0.524626
$161$	−2.02692	−0.159743
$162$	0	0
$163$	9.89229	0.774824	0.387412	−	0.921907i	$-0.373369\pi$
$163$	9.89229	0.774824	0.387412	+	0.921907i	$0.373369\pi$
$164$	6.51231	0.508526
$165$	0	0
$166$	2.24338	0.174120
$167$	−7.71047	−0.596654	−0.298327	−	0.954464i	$-0.596429\pi$
$167$	−7.71047	−0.596654	−0.298327	+	0.954464i	$0.596429\pi$
$168$	0	0
$169$	17.8423	1.37249
$170$	2.30644	0.176896
$171$	0	0
$172$	4.67067	0.356135
$173$	−13.4127	−1.01975	−0.509873	−	0.860250i	$-0.670308\pi$
$173$	−13.4127	−1.01975	−0.509873	+	0.860250i	$0.670308\pi$
$174$	0	0
$175$	−0.978959	−0.0740023
$176$	0.130935	0.00986961
$177$	0	0
$178$	−3.00594	−0.225305
$179$	−6.22605	−0.465357	−0.232678	−	0.972554i	$-0.574749\pi$
$179$	−6.22605	−0.465357	−0.232678	+	0.972554i	$0.574749\pi$
$180$	0	0
$181$	−19.9982	−1.48645	−0.743226	−	0.669041i	$-0.766705\pi$
$181$	−19.9982	−1.48645	−0.743226	+	0.669041i	$0.766705\pi$
$182$	−1.03567	−0.0767691
$183$	0	0
$184$	2.26980	0.167332
$185$	17.7647	1.30608
$186$	0	0
$187$	−0.142988	−0.0104563
$188$	1.94709	0.142006
$189$	0	0
$190$	0.573085	0.0415759
$191$	13.3880	0.968719	0.484360	−	0.874869i	$-0.339053\pi$
$191$	13.3880	0.968719	0.484360	+	0.874869i	$0.339053\pi$
$192$	0	0
$193$	12.1130	0.871910	0.435955	−	0.899968i	$-0.356411\pi$
$193$	12.1130	0.871910	0.435955	+	0.899968i	$0.356411\pi$
$194$	0.0896301	0.00643507
$195$	0	0
$196$	12.3498	0.882128
$197$	14.8880	1.06073	0.530364	−	0.847770i	$-0.322055\pi$
$197$	14.8880	1.06073	0.530364	+	0.847770i	$0.322055\pi$
$198$	0	0
$199$	5.79677	0.410922	0.205461	−	0.978665i	$-0.434131\pi$
$199$	5.79677	0.410922	0.205461	+	0.978665i	$0.434131\pi$
$200$	1.09627	0.0775177
$201$	0	0
$202$	−1.39613	−0.0982316
$203$	3.91598	0.274848
$204$	0	0
$205$	−8.33309	−0.582008
$206$	0.830784	0.0578834
$207$	0	0
$208$	−20.4669	−1.41912
$209$	−0.0355286	−0.00245756
$210$	0	0
$211$	16.6533	1.14646	0.573231	−	0.819394i	$-0.305690\pi$
$211$	16.6533	1.14646	0.573231	+	0.819394i	$0.305690\pi$
$212$	−14.6153	−1.00378
$213$	0	0
$214$	−1.24447	−0.0850701
$215$	−5.97655	−0.407597
$216$	0	0
$217$	1.85430	0.125878
$218$	−1.48366	−0.100486
$219$	0	0
$220$	−0.172354	−0.0116201
$221$	22.3510	1.50349
$222$	0	0
$223$	8.90873	0.596573	0.298286	−	0.954476i	$-0.403585\pi$
$223$	8.90873	0.596573	0.298286	+	0.954476i	$0.403585\pi$
$224$	2.15943	0.144283
$225$	0	0
$226$	−1.81072	−0.120447
$227$	−21.2720	−1.41187	−0.705937	−	0.708275i	$-0.749474\pi$
$227$	−21.2720	−1.41187	−0.705937	+	0.708275i	$0.749474\pi$
$228$	0	0
$229$	5.46030	0.360827	0.180413	−	0.983591i	$-0.442256\pi$
$229$	5.46030	0.360827	0.180413	+	0.983591i	$0.442256\pi$
$230$	−1.43274	−0.0944722
$231$	0	0
$232$	−4.38522	−0.287904
$233$	−13.2517	−0.868147	−0.434074	−	0.900877i	$-0.642924\pi$
$233$	−13.2517	−0.868147	−0.434074	+	0.900877i	$0.642924\pi$
$234$	0	0
$235$	−2.49148	−0.162526
$236$	1.48964	0.0969672
$237$	0	0
$238$	−0.750536	−0.0486500
$239$	−1.07041	−0.0692394	−0.0346197	−	0.999401i	$-0.511022\pi$
$239$	−1.07041	−0.0692394	−0.0346197	+	0.999401i	$0.511022\pi$
$240$	0	0
$241$	17.5729	1.13197	0.565985	−	0.824415i	$-0.308496\pi$
$241$	17.5729	1.13197	0.565985	+	0.824415i	$0.308496\pi$
$242$	2.52991	0.162628
$243$	0	0
$244$	−17.8666	−1.14379
$245$	−15.8027	−1.00960
$246$	0	0
$247$	5.55359	0.353366
$248$	−2.07650	−0.131858
$249$	0	0
$250$	2.17344	0.137460
$251$	−14.6184	−0.922703	−0.461351	−	0.887218i	$-0.652635\pi$
$251$	−14.6184	−0.922703	−0.461351	+	0.887218i	$0.652635\pi$
$252$	0	0
$253$	0.0888232	0.00558427
$254$	2.82779	0.177431
$255$	0	0
$256$	11.9332	0.745827
$257$	1.67160	0.104272	0.0521358	−	0.998640i	$-0.483397\pi$
$257$	1.67160	0.104272	0.0521358	+	0.998640i	$0.483397\pi$
$258$	0	0
$259$	−5.78079	−0.359201
$260$	26.9412	1.67082
$261$	0	0
$262$	−1.85876	−0.114835
$263$	−12.9805	−0.800410	−0.400205	−	0.916426i	$-0.631061\pi$
$263$	−12.9805	−0.800410	−0.400205	+	0.916426i	$0.631061\pi$
$264$	0	0
$265$	18.7016	1.14883
$266$	−0.186487	−0.0114343
$267$	0	0
$268$	−15.4024	−0.940850
$269$	4.91538	0.299696	0.149848	−	0.988709i	$-0.452122\pi$
$269$	4.91538	0.299696	0.149848	+	0.988709i	$0.452122\pi$
$270$	0	0
$271$	−7.09560	−0.431027	−0.215514	−	0.976501i	$-0.569143\pi$
$271$	−7.09560	−0.431027	−0.215514	+	0.976501i	$0.569143\pi$
$272$	−14.8321	−0.899326
$273$	0	0
$274$	−1.73896	−0.105055
$275$	0.0428998	0.00258695
$276$	0	0
$277$	−11.9075	−0.715453	−0.357726	−	0.933826i	$-0.616448\pi$
$277$	−11.9075	−0.715453	−0.357726	+	0.933826i	$0.616448\pi$
$278$	−4.71201	−0.282608
$279$	0	0
$280$	−1.83393	−0.109598
$281$	9.41327	0.561549	0.280774	−	0.959774i	$-0.409409\pi$
$281$	9.41327	0.561549	0.280774	+	0.959774i	$0.409409\pi$
$282$	0	0
$283$	13.1197	0.779885	0.389942	−	0.920839i	$-0.372495\pi$
$283$	13.1197	0.779885	0.389942	+	0.920839i	$0.372495\pi$
$284$	−5.63728	−0.334511
$285$	0	0
$286$	0.0453850	0.00268367
$287$	2.71166	0.160065
$288$	0	0
$289$	−0.802589	−0.0472111
$290$	2.76803	0.162545
$291$	0	0
$292$	−26.5487	−1.55364
$293$	11.9102	0.695803	0.347901	−	0.937531i	$-0.386894\pi$
$293$	11.9102	0.695803	0.347901	+	0.937531i	$0.386894\pi$
$294$	0	0
$295$	−1.90613	−0.110979
$296$	6.47349	0.376264
$297$	0	0
$298$	1.06523	0.0617069
$299$	−13.8842	−0.802947
$300$	0	0
$301$	1.94482	0.112098
$302$	−3.47854	−0.200167
$303$	0	0
$304$	−3.68535	−0.211369
$305$	22.8619	1.30907
$306$	0	0
$307$	22.4399	1.28071	0.640355	−	0.768079i	$-0.278787\pi$
$307$	22.4399	1.28071	0.640355	+	0.768079i	$0.278787\pi$
$308$	0.0560856	0.00319577
$309$	0	0
$310$	1.31073	0.0744442
$311$	21.1277	1.19804	0.599021	−	0.800733i	$-0.295556\pi$
$311$	21.1277	1.19804	0.599021	+	0.800733i	$0.295556\pi$
$312$	0	0
$313$	−2.78495	−0.157415	−0.0787074	−	0.996898i	$-0.525079\pi$
$313$	−2.78495	−0.157415	−0.0787074	+	0.996898i	$0.525079\pi$
$314$	−0.277752	−0.0156745
$315$	0	0
$316$	6.09367	0.342796
$317$	−10.5410	−0.592044	−0.296022	−	0.955181i	$-0.595660\pi$
$317$	−10.5410	−0.592044	−0.296022	+	0.955181i	$0.595660\pi$
$318$	0	0
$319$	−0.171605	−0.00960804
$320$	−16.8375	−0.941247
$321$	0	0
$322$	0.466227	0.0259818
$323$	4.02460	0.223935
$324$	0	0
$325$	−6.70580	−0.371971
$326$	−2.27540	−0.126023
$327$	0	0
$328$	−3.03660	−0.167668
$329$	0.810750	0.0446981
$330$	0	0
$331$	22.9162	1.25959	0.629794	−	0.776762i	$-0.283139\pi$
$331$	22.9162	1.25959	0.629794	+	0.776762i	$0.283139\pi$
$332$	18.9902	1.04222
$333$	0	0
$334$	1.77354	0.0970441
$335$	19.7087	1.07680
$336$	0	0
$337$	−32.7916	−1.78627	−0.893137	−	0.449784i	$-0.851501\pi$
$337$	−32.7916	−1.78627	−0.893137	+	0.449784i	$0.851501\pi$
$338$	−4.10405	−0.223231
$339$	0	0
$340$	19.5239	1.05883
$341$	−0.0812588	−0.00440041
$342$	0	0
$343$	10.8176	0.584095
$344$	−2.17787	−0.117423
$345$	0	0
$346$	3.08515	0.165859
$347$	−24.7745	−1.32996	−0.664981	−	0.746860i	$-0.731561\pi$
$347$	−24.7745	−1.32996	−0.664981	+	0.746860i	$0.731561\pi$
$348$	0	0
$349$	−34.4877	−1.84609	−0.923043	−	0.384696i	$-0.874306\pi$
$349$	−34.4877	−1.84609	−0.923043	+	0.384696i	$0.874306\pi$
$350$	0.225178	0.0120363
$351$	0	0
$352$	−0.0946303	−0.00504381
$353$	−0.916653	−0.0487885	−0.0243943	−	0.999702i	$-0.507766\pi$
$353$	−0.916653	−0.0487885	−0.0243943	+	0.999702i	$0.507766\pi$
$354$	0	0
$355$	7.21340	0.382848
$356$	−25.4452	−1.34859
$357$	0	0
$358$	1.43210	0.0756890
$359$	−18.7717	−0.990731	−0.495366	−	0.868685i	$-0.664966\pi$
$359$	−18.7717	−0.990731	−0.495366	+	0.868685i	$0.664966\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	4.59993	0.241767
$363$	0	0
$364$	−8.76692	−0.459511
$365$	33.9714	1.77815
$366$	0	0
$367$	36.5340	1.90706	0.953529	−	0.301300i	$-0.0974207\pi$
$367$	36.5340	1.90706	0.953529	+	0.301300i	$0.0974207\pi$
$368$	9.21356	0.480290
$369$	0	0
$370$	−4.08619	−0.212431
$371$	−6.08566	−0.315952
$372$	0	0
$373$	28.0962	1.45477	0.727384	−	0.686231i	$-0.240736\pi$
$373$	28.0962	1.45477	0.727384	+	0.686231i	$0.240736\pi$
$374$	0.0328898	0.00170069
$375$	0	0
$376$	−0.907901	−0.0468214
$377$	26.8242	1.38151
$378$	0	0
$379$	2.30867	0.118588	0.0592942	−	0.998241i	$-0.481115\pi$
$379$	2.30867	0.118588	0.0592942	+	0.998241i	$0.481115\pi$
$380$	4.85114	0.248858
$381$	0	0
$382$	−3.07947	−0.157559
$383$	−1.08845	−0.0556173	−0.0278087	−	0.999613i	$-0.508853\pi$
$383$	−1.08845	−0.0556173	−0.0278087	+	0.999613i	$0.508853\pi$
$384$	0	0
$385$	−0.0717666	−0.00365756
$386$	−2.78620	−0.141814
$387$	0	0
$388$	0.758715	0.0385179
$389$	−18.8780	−0.957151	−0.478576	−	0.878046i	$-0.658847\pi$
$389$	−18.8780	−0.957151	−0.478576	+	0.878046i	$0.658847\pi$
$390$	0	0
$391$	−10.0617	−0.508842
$392$	−5.75853	−0.290850
$393$	0	0
$394$	−3.42451	−0.172524
$395$	−7.79740	−0.392330
$396$	0	0
$397$	6.58403	0.330443	0.165221	−	0.986256i	$-0.447166\pi$
$397$	6.58403	0.330443	0.165221	+	0.986256i	$0.447166\pi$
$398$	−1.33336	−0.0668353
$399$	0	0
$400$	4.44996	0.222498
$401$	0.362406	0.0180977	0.00904885	−	0.999959i	$-0.497120\pi$
$401$	0.362406	0.0180977	0.00904885	+	0.999959i	$0.497120\pi$
$402$	0	0
$403$	12.7018	0.632723
$404$	−11.8182	−0.587978
$405$	0	0
$406$	−0.900744	−0.0447032
$407$	0.253325	0.0125568
$408$	0	0
$409$	12.4155	0.613905	0.306953	−	0.951725i	$-0.400691\pi$
$409$	12.4155	0.613905	0.306953	+	0.951725i	$0.400691\pi$
$410$	1.91676	0.0946620
$411$	0	0
$412$	7.03255	0.346469
$413$	0.620271	0.0305216
$414$	0	0
$415$	−24.2996	−1.19282
$416$	14.7920	0.725236
$417$	0	0
$418$	0.00817220	0.000399715 0
$419$	−29.9200	−1.46169	−0.730843	−	0.682546i	$-0.760873\pi$
$419$	−29.9200	−1.46169	−0.730843	+	0.682546i	$0.760873\pi$
$420$	0	0
$421$	20.6200	1.00496	0.502479	−	0.864589i	$-0.332421\pi$
$421$	20.6200	1.00496	0.502479	+	0.864589i	$0.332421\pi$
$422$	−3.83056	−0.186469
$423$	0	0
$424$	6.81489	0.330960
$425$	−4.85960	−0.235725
$426$	0	0
$427$	−7.43949	−0.360022
$428$	−10.5344	−0.509198
$429$	0	0
$430$	1.37471	0.0662945
$431$	29.3142	1.41202	0.706008	−	0.708204i	$-0.250494\pi$
$431$	29.3142	1.41202	0.706008	+	0.708204i	$0.250494\pi$
$432$	0	0
$433$	35.2087	1.69202	0.846010	−	0.533167i	$-0.178998\pi$
$433$	35.2087	1.69202	0.846010	+	0.533167i	$0.178998\pi$
$434$	−0.426522	−0.0204737
$435$	0	0
$436$	−12.5591	−0.601473
$437$	−2.50005	−0.119594
$438$	0	0
$439$	31.2216	1.49013	0.745063	−	0.666994i	$-0.232419\pi$
$439$	31.2216	1.49013	0.745063	+	0.666994i	$0.232419\pi$
$440$	0.0803663	0.00383131
$441$	0	0
$442$	−5.14112	−0.244538
$443$	1.65785	0.0787671	0.0393835	−	0.999224i	$-0.487461\pi$
$443$	1.65785	0.0787671	0.0393835	+	0.999224i	$0.487461\pi$
$444$	0	0
$445$	32.5594	1.54346
$446$	−2.04917	−0.0970309
$447$	0	0
$448$	5.47909	0.258863
$449$	13.2151	0.623659	0.311830	−	0.950138i	$-0.399058\pi$
$449$	13.2151	0.623659	0.311830	+	0.950138i	$0.399058\pi$
$450$	0	0
$451$	−0.118830	−0.00559549
$452$	−15.3277	−0.720954
$453$	0	0
$454$	4.89294	0.229637
$455$	11.2181	0.525911
$456$	0	0
$457$	3.07243	0.143722	0.0718611	−	0.997415i	$-0.477106\pi$
$457$	3.07243	0.143722	0.0718611	+	0.997415i	$0.477106\pi$
$458$	−1.25597	−0.0586874
$459$	0	0
$460$	−12.1281	−0.565476
$461$	6.06028	0.282255	0.141128	−	0.989991i	$-0.454927\pi$
$461$	6.06028	0.282255	0.141128	+	0.989991i	$0.454927\pi$
$462$	0	0
$463$	−28.6953	−1.33358	−0.666792	−	0.745244i	$-0.732333\pi$
$463$	−28.6953	−1.33358	−0.666792	+	0.745244i	$0.732333\pi$
$464$	−17.8005	−0.826366
$465$	0	0
$466$	3.04812	0.141202
$467$	13.6640	0.632296	0.316148	−	0.948710i	$-0.397610\pi$
$467$	13.6640	0.632296	0.316148	+	0.948710i	$0.397610\pi$
$468$	0	0
$469$	−6.41340	−0.296143
$470$	0.573085	0.0264344
$471$	0	0
$472$	−0.694598	−0.0319714
$473$	−0.0852257	−0.00391868
$474$	0	0
$475$	−1.20747	−0.0554026
$476$	−6.35326	−0.291201
$477$	0	0
$478$	0.246214	0.0112616
$479$	18.9827	0.867343	0.433671	−	0.901071i	$-0.357218\pi$
$479$	18.9827	0.867343	0.433671	+	0.901071i	$0.357218\pi$
$480$	0	0
$481$	−39.5980	−1.80551
$482$	−4.04208	−0.184112
$483$	0	0
$484$	21.4156	0.973434
$485$	−0.970845	−0.0440838
$486$	0	0
$487$	31.9145	1.44618	0.723091	−	0.690752i	$-0.242721\pi$
$487$	31.9145	1.44618	0.723091	+	0.690752i	$0.242721\pi$
$488$	8.33095	0.377124
$489$	0	0
$490$	3.63490	0.164208
$491$	0.396011	0.0178717	0.00893586	−	0.999960i	$-0.497156\pi$
$491$	0.396011	0.0178717	0.00893586	+	0.999960i	$0.497156\pi$
$492$	0	0
$493$	19.4391	0.875492
$494$	−1.27742	−0.0574740
$495$	0	0
$496$	−8.42892	−0.378469
$497$	−2.34731	−0.105291
$498$	0	0
$499$	−36.1035	−1.61622	−0.808108	−	0.589035i	$-0.799508\pi$
$499$	−36.1035	−1.61622	−0.808108	+	0.589035i	$0.799508\pi$
$500$	18.3981	0.822787
$501$	0	0
$502$	3.36248	0.150075
$503$	33.7712	1.50578	0.752892	−	0.658144i	$-0.228658\pi$
$503$	33.7712	1.50578	0.752892	+	0.658144i	$0.228658\pi$
$504$	0	0
$505$	15.1225	0.672941
$506$	−0.0204309	−0.000908265 0
$507$	0	0
$508$	23.9371	1.06204
$509$	36.6836	1.62597	0.812986	−	0.582283i	$-0.197841\pi$
$509$	36.6836	1.62597	0.812986	+	0.582283i	$0.197841\pi$
$510$	0	0
$511$	−11.0546	−0.489027
$512$	−16.5078	−0.729548
$513$	0	0
$514$	−0.384498	−0.0169595
$515$	−8.99879	−0.396534
$516$	0	0
$517$	−0.0355286	−0.00156254
$518$	1.32968	0.0584230
$519$	0	0
$520$	−12.5623	−0.550894
$521$	−40.3436	−1.76749	−0.883743	−	0.467972i	$-0.844985\pi$
$521$	−40.3436	−1.76749	−0.883743	+	0.467972i	$0.844985\pi$
$522$	0	0
$523$	−7.02835	−0.307328	−0.153664	−	0.988123i	$-0.549107\pi$
$523$	−7.02835	−0.307328	−0.153664	+	0.988123i	$0.549107\pi$
$524$	−15.7343	−0.687358
$525$	0	0
$526$	2.98574	0.130184
$527$	9.20483	0.400969
$528$	0	0
$529$	−16.7497	−0.728250
$530$	−4.30169	−0.186854
$531$	0	0
$532$	−1.57861	−0.0684412
$533$	18.5747	0.804560
$534$	0	0
$535$	13.4797	0.582777
$536$	7.18191	0.310211
$537$	0	0
$538$	−1.13063	−0.0487447
$539$	−0.225346	−0.00970636
$540$	0	0
$541$	35.9200	1.54432	0.772161	−	0.635427i	$-0.219176\pi$
$541$	35.9200	1.54432	0.772161	+	0.635427i	$0.219176\pi$
$542$	1.63212	0.0701054
$543$	0	0
$544$	10.7195	0.459596
$545$	16.0705	0.688386
$546$	0	0
$547$	21.3410	0.912477	0.456238	−	0.889858i	$-0.349196\pi$
$547$	21.3410	0.912477	0.456238	+	0.889858i	$0.349196\pi$
$548$	−14.7203	−0.628819
$549$	0	0
$550$	−0.00986771	−0.000420761 0
$551$	4.83006	0.205768
$552$	0	0
$553$	2.53734	0.107899
$554$	2.73894	0.116366
$555$	0	0
$556$	−39.8870	−1.69159
$557$	23.7346	1.00567	0.502833	−	0.864384i	$-0.332291\pi$
$557$	23.7346	1.00567	0.502833	+	0.864384i	$0.332291\pi$
$558$	0	0
$559$	13.3219	0.563457
$560$	−7.44429	−0.314579
$561$	0	0
$562$	−2.16522	−0.0913343
$563$	−27.8870	−1.17530	−0.587648	−	0.809117i	$-0.699946\pi$
$563$	−27.8870	−1.17530	−0.587648	+	0.809117i	$0.699946\pi$
$564$	0	0
$565$	19.6132	0.825132
$566$	−3.01776	−0.126846
$567$	0	0
$568$	2.62858	0.110293
$569$	7.84084	0.328705	0.164353	−	0.986402i	$-0.447447\pi$
$569$	7.84084	0.328705	0.164353	+	0.986402i	$0.447447\pi$
$570$	0	0
$571$	4.35696	0.182333	0.0911666	−	0.995836i	$-0.470940\pi$
$571$	4.35696	0.182333	0.0911666	+	0.995836i	$0.470940\pi$
$572$	0.384182	0.0160635
$573$	0	0
$574$	−0.623731	−0.0260340
$575$	3.01874	0.125890
$576$	0	0
$577$	30.9022	1.28648	0.643238	−	0.765667i	$-0.277591\pi$
$577$	30.9022	1.28648	0.643238	+	0.765667i	$0.277591\pi$
$578$	0.184610	0.00767875
$579$	0	0
$580$	23.4313	0.972932
$581$	7.90732	0.328051
$582$	0	0
$583$	0.266685	0.0110449
$584$	12.3793	0.512258
$585$	0	0
$586$	−2.73956	−0.113170
$587$	−19.5632	−0.807460	−0.403730	−	0.914878i	$-0.632286\pi$
$587$	−19.5632	−0.807460	−0.403730	+	0.914878i	$0.632286\pi$
$588$	0	0
$589$	2.28714	0.0942400
$590$	0.438443	0.0180504
$591$	0	0
$592$	26.2772	1.07998
$593$	32.7215	1.34371	0.671856	−	0.740682i	$-0.265497\pi$
$593$	32.7215	1.34371	0.671856	+	0.740682i	$0.265497\pi$
$594$	0	0
$595$	8.12957	0.333280
$596$	9.01709	0.369355
$597$	0	0
$598$	3.19362	0.130597
$599$	−17.0660	−0.697299	−0.348649	−	0.937253i	$-0.613360\pi$
$599$	−17.0660	−0.697299	−0.348649	+	0.937253i	$0.613360\pi$
$600$	0	0
$601$	15.3178	0.624828	0.312414	−	0.949946i	$-0.398862\pi$
$601$	15.3178	0.624828	0.312414	+	0.949946i	$0.398862\pi$
$602$	−0.447344	−0.0182324
$603$	0	0
$604$	−29.4457	−1.19813
$605$	−27.4031	−1.11410
$606$	0	0
$607$	28.8270	1.17005	0.585025	−	0.811015i	$-0.301085\pi$
$607$	28.8270	1.17005	0.585025	+	0.811015i	$0.301085\pi$
$608$	2.66350	0.108019
$609$	0	0
$610$	−5.25865	−0.212917
$611$	5.55359	0.224674
$612$	0	0
$613$	−31.1850	−1.25955	−0.629775	−	0.776778i	$-0.716853\pi$
$613$	−31.1850	−1.25955	−0.629775	+	0.776778i	$0.716853\pi$
$614$	−5.16157	−0.208304
$615$	0	0
$616$	−0.0261519	−0.00105369
$617$	7.74695	0.311881	0.155940	−	0.987766i	$-0.450159\pi$
$617$	7.74695	0.311881	0.155940	+	0.987766i	$0.450159\pi$
$618$	0	0
$619$	10.9903	0.441737	0.220869	−	0.975304i	$-0.429111\pi$
$619$	10.9903	0.441737	0.220869	+	0.975304i	$0.429111\pi$
$620$	11.0952	0.445596
$621$	0	0
$622$	−4.85975	−0.194858
$623$	−10.5951	−0.424485
$624$	0	0
$625$	−29.5794	−1.18317
$626$	0.640588	0.0256031
$627$	0	0
$628$	−2.35116	−0.0938216
$629$	−28.6961	−1.14419
$630$	0	0
$631$	−4.49783	−0.179056	−0.0895279	−	0.995984i	$-0.528536\pi$
$631$	−4.49783	−0.179056	−0.0895279	+	0.995984i	$0.528536\pi$
$632$	−2.84139	−0.113024
$633$	0	0
$634$	2.42463	0.0962942
$635$	−30.6297	−1.21550
$636$	0	0
$637$	35.2246	1.39565
$638$	0.0394722	0.00156272
$639$	0	0
$640$	17.1450	0.677717
$641$	1.48238	0.0585504	0.0292752	−	0.999571i	$-0.490680\pi$
$641$	1.48238	0.0585504	0.0292752	+	0.999571i	$0.490680\pi$
$642$	0	0
$643$	4.83557	0.190696	0.0953481	−	0.995444i	$-0.469604\pi$
$643$	4.83557	0.190696	0.0953481	+	0.995444i	$0.469604\pi$
$644$	3.94659	0.155518
$645$	0	0
$646$	−0.925730	−0.0364224
$647$	−22.1893	−0.872350	−0.436175	−	0.899862i	$-0.643667\pi$
$647$	−22.1893	−0.872350	−0.436175	+	0.899862i	$0.643667\pi$
$648$	0	0
$649$	−0.0271814	−0.00106696
$650$	1.54245	0.0605000
$651$	0	0
$652$	−19.2612	−0.754327
$653$	36.9965	1.44778	0.723892	−	0.689914i	$-0.242352\pi$
$653$	36.9965	1.44778	0.723892	+	0.689914i	$0.242352\pi$
$654$	0	0
$655$	20.1335	0.786682
$656$	−12.3262	−0.481255
$657$	0	0
$658$	−0.186487	−0.00727002
$659$	39.9834	1.55753	0.778766	−	0.627314i	$-0.215846\pi$
$659$	39.9834	1.55753	0.778766	+	0.627314i	$0.215846\pi$
$660$	0	0
$661$	−15.7367	−0.612088	−0.306044	−	0.952017i	$-0.599005\pi$
$661$	−15.7367	−0.612088	−0.306044	+	0.952017i	$0.599005\pi$
$662$	−5.27113	−0.204868
$663$	0	0
$664$	−8.85484	−0.343634
$665$	2.01997	0.0783310
$666$	0	0
$667$	−12.0754	−0.467561
$668$	15.0130	0.580870
$669$	0	0
$670$	−4.53336	−0.175139
$671$	0.326012	0.0125855
$672$	0	0
$673$	−2.85642	−0.110107	−0.0550535	−	0.998483i	$-0.517533\pi$
$673$	−2.85642	−0.110107	−0.0550535	+	0.998483i	$0.517533\pi$
$674$	7.54266	0.290532
$675$	0	0
$676$	−34.7406	−1.33618
$677$	−50.3250	−1.93415	−0.967073	−	0.254498i	$-0.918090\pi$
$677$	−50.3250	−1.93415	−0.967073	+	0.254498i	$0.918090\pi$
$678$	0	0
$679$	0.315922	0.0121240
$680$	−9.10372	−0.349112
$681$	0	0
$682$	0.0186910	0.000715715 0
$683$	43.3148	1.65740	0.828698	−	0.559696i	$-0.189082\pi$
$683$	43.3148	1.65740	0.828698	+	0.559696i	$0.189082\pi$
$684$	0	0
$685$	18.8359	0.719683
$686$	−2.48824	−0.0950013
$687$	0	0
$688$	−8.84039	−0.337037
$689$	−41.6863	−1.58812
$690$	0	0
$691$	−32.0989	−1.22110	−0.610549	−	0.791978i	$-0.709051\pi$
$691$	−32.0989	−1.22110	−0.610549	+	0.791978i	$0.709051\pi$
$692$	26.1157	0.992769
$693$	0	0
$694$	5.69857	0.216315
$695$	51.0390	1.93602
$696$	0	0
$697$	13.4608	0.509865
$698$	7.93280	0.300261
$699$	0	0
$700$	1.90612	0.0720447
$701$	8.56683	0.323565	0.161782	−	0.986826i	$-0.448276\pi$
$701$	8.56683	0.323565	0.161782	+	0.986826i	$0.448276\pi$
$702$	0	0
$703$	−7.13017	−0.268919
$704$	−0.240104	−0.00904925
$705$	0	0
$706$	0.210846	0.00793531
$707$	−4.92099	−0.185073
$708$	0	0
$709$	−31.2997	−1.17548	−0.587742	−	0.809048i	$-0.699983\pi$
$709$	−31.2997	−1.17548	−0.587742	+	0.809048i	$0.699983\pi$
$710$	−1.65921	−0.0622691
$711$	0	0
$712$	11.8647	0.444649
$713$	−5.71797	−0.214140
$714$	0	0
$715$	−0.491596	−0.0183847
$716$	12.1227	0.453046
$717$	0	0
$718$	4.31782	0.161140
$719$	50.5470	1.88509	0.942543	−	0.334086i	$-0.108428\pi$
$719$	50.5470	1.88509	0.942543	+	0.334086i	$0.108428\pi$
$720$	0	0
$721$	2.92829	0.109055
$722$	−0.230018	−0.00856038
$723$	0	0
$724$	38.9383	1.44713
$725$	−5.83217	−0.216601
$726$	0	0
$727$	40.6579	1.50792	0.753959	−	0.656922i	$-0.228142\pi$
$727$	40.6579	1.50792	0.753959	+	0.656922i	$0.228142\pi$
$728$	4.08789	0.151507
$729$	0	0
$730$	−7.81403	−0.289210
$731$	9.65419	0.357073
$732$	0	0
$733$	24.0407	0.887964	0.443982	−	0.896036i	$-0.353565\pi$
$733$	24.0407	0.887964	0.443982	+	0.896036i	$0.353565\pi$
$734$	−8.40347	−0.310178
$735$	0	0
$736$	−6.65888	−0.245450
$737$	0.281047	0.0103525
$738$	0	0
$739$	24.7944	0.912075	0.456038	−	0.889961i	$-0.349268\pi$
$739$	24.7944	0.912075	0.456038	+	0.889961i	$0.349268\pi$
$740$	−34.5894	−1.27153
$741$	0	0
$742$	1.39981	0.0513886
$743$	−25.1853	−0.923959	−0.461979	−	0.886891i	$-0.652861\pi$
$743$	−25.1853	−0.923959	−0.461979	+	0.886891i	$0.652861\pi$
$744$	0	0
$745$	−11.5382	−0.422726
$746$	−6.46263	−0.236614
$747$	0	0
$748$	0.278411	0.0101797
$749$	−4.38641	−0.160276
$750$	0	0
$751$	−25.8609	−0.943678	−0.471839	−	0.881685i	$-0.656410\pi$
$751$	−25.8609	−0.943678	−0.471839	+	0.881685i	$0.656410\pi$
$752$	−3.68535	−0.134391
$753$	0	0
$754$	−6.17003	−0.224699
$755$	37.6784	1.37126
$756$	0	0
$757$	2.76180	0.100379	0.0501897	−	0.998740i	$-0.484017\pi$
$757$	2.76180	0.100379	0.0501897	+	0.998740i	$0.484017\pi$
$758$	−0.531035	−0.0192881
$759$	0	0
$760$	−2.26202	−0.0820520
$761$	43.5674	1.57932	0.789658	−	0.613547i	$-0.210258\pi$
$761$	43.5674	1.57932	0.789658	+	0.613547i	$0.210258\pi$
$762$	0	0
$763$	−5.22950	−0.189321
$764$	−26.0676	−0.943093
$765$	0	0
$766$	0.250364	0.00904600
$767$	4.24882	0.153416
$768$	0	0
$769$	−19.4331	−0.700774	−0.350387	−	0.936605i	$-0.613950\pi$
$769$	−19.4331	−0.700774	−0.350387	+	0.936605i	$0.613950\pi$
$770$	0.0165076	0.000594892 0
$771$	0	0
$772$	−23.5851	−0.848845
$773$	−29.0184	−1.04372	−0.521860	−	0.853031i	$-0.674762\pi$
$773$	−29.0184	−1.04372	−0.521860	+	0.853031i	$0.674762\pi$
$774$	0	0
$775$	−2.76166	−0.0992018
$776$	−0.353778	−0.0126999
$777$	0	0
$778$	4.34227	0.155678
$779$	3.34464	0.119834
$780$	0	0
$781$	0.102863	0.00368074
$782$	2.31437	0.0827618
$783$	0	0
$784$	−23.3750	−0.834822
$785$	3.00852	0.107379
$786$	0	0
$787$	33.6161	1.19828	0.599142	−	0.800643i	$-0.295509\pi$
$787$	33.6161	1.19828	0.599142	+	0.800643i	$0.295509\pi$
$788$	−28.9883	−1.03267
$789$	0	0
$790$	1.79354	0.0638113
$791$	−6.38230	−0.226929
$792$	0	0
$793$	−50.9600	−1.80964
$794$	−1.51444	−0.0537456
$795$	0	0
$796$	−11.2868	−0.400052
$797$	44.7014	1.58341	0.791703	−	0.610906i	$-0.209195\pi$
$797$	44.7014	1.58341	0.791703	+	0.610906i	$0.209195\pi$
$798$	0	0
$799$	4.02460	0.142380
$800$	−3.21610	−0.113706
$801$	0	0
$802$	−0.0833598	−0.00294354
$803$	0.484433	0.0170953
$804$	0	0
$805$	−5.05002	−0.177990
$806$	−2.92165	−0.102911
$807$	0	0
$808$	5.51067	0.193865
$809$	−38.8162	−1.36471	−0.682353	−	0.731023i	$-0.739043\pi$
$809$	−38.8162	−1.36471	−0.682353	+	0.731023i	$0.739043\pi$
$810$	0	0
$811$	24.3086	0.853590	0.426795	−	0.904348i	$-0.359643\pi$
$811$	24.3086	0.853590	0.426795	+	0.904348i	$0.359643\pi$
$812$	−7.62476	−0.267577
$813$	0	0
$814$	−0.0582692	−0.00204233
$815$	24.6464	0.863327
$816$	0	0
$817$	2.39879	0.0839232
$818$	−2.85578	−0.0998499
$819$	0	0
$820$	16.2253	0.566612
$821$	6.37891	0.222626	0.111313	−	0.993785i	$-0.464494\pi$
$821$	6.37891	0.222626	0.111313	+	0.993785i	$0.464494\pi$
$822$	0	0
$823$	13.9570	0.486512	0.243256	−	0.969962i	$-0.421784\pi$
$823$	13.9570	0.486512	0.243256	+	0.969962i	$0.421784\pi$
$824$	−3.27918	−0.114236
$825$	0	0
$826$	−0.142673	−0.00496425
$827$	−9.65578	−0.335764	−0.167882	−	0.985807i	$-0.553693\pi$
$827$	−9.65578	−0.335764	−0.167882	+	0.985807i	$0.553693\pi$
$828$	0	0
$829$	−0.430277	−0.0149441	−0.00747207	−	0.999972i	$-0.502378\pi$
$829$	−0.430277	−0.0149441	−0.00747207	+	0.999972i	$0.502378\pi$
$830$	5.58935	0.194009
$831$	0	0
$832$	37.5314	1.30117
$833$	25.5268	0.884450
$834$	0	0
$835$	−19.2105	−0.664806
$836$	0.0691774	0.00239255
$837$	0	0
$838$	6.88212	0.237739
$839$	45.6446	1.57583	0.787913	−	0.615787i	$-0.211162\pi$
$839$	45.6446	1.57583	0.787913	+	0.615787i	$0.211162\pi$
$840$	0	0
$841$	−5.67050	−0.195534
$842$	−4.74297	−0.163454
$843$	0	0
$844$	−32.4256	−1.11613
$845$	44.4538	1.52926
$846$	0	0
$847$	8.91723	0.306400
$848$	27.6630	0.949951
$849$	0	0
$850$	1.11779	0.0383400
$851$	17.8258	0.611060
$852$	0	0
$853$	3.94406	0.135042	0.0675210	−	0.997718i	$-0.478491\pi$
$853$	3.94406	0.135042	0.0675210	+	0.997718i	$0.478491\pi$
$854$	1.71121	0.0585566
$855$	0	0
$856$	4.91203	0.167890
$857$	−30.1814	−1.03098	−0.515488	−	0.856897i	$-0.672389\pi$
$857$	−30.1814	−1.03098	−0.515488	+	0.856897i	$0.672389\pi$
$858$	0	0
$859$	−31.5115	−1.07516	−0.537580	−	0.843213i	$-0.680661\pi$
$859$	−31.5115	−1.07516	−0.537580	+	0.843213i	$0.680661\pi$
$860$	11.6369	0.396814
$861$	0	0
$862$	−6.74279	−0.229660
$863$	10.1978	0.347139	0.173569	−	0.984822i	$-0.444470\pi$
$863$	10.1978	0.347139	0.173569	+	0.984822i	$0.444470\pi$
$864$	0	0
$865$	−33.4174	−1.13622
$866$	−8.09862	−0.275202
$867$	0	0
$868$	−3.61049	−0.122548
$869$	−0.111191	−0.00377190
$870$	0	0
$871$	−43.9314	−1.48856
$872$	5.85614	0.198314
$873$	0	0
$874$	0.575056	0.0194516
$875$	7.66079	0.258982
$876$	0	0
$877$	−44.0592	−1.48777	−0.743886	−	0.668306i	$-0.767020\pi$
$877$	−44.0592	−1.48777	−0.743886	+	0.668306i	$0.767020\pi$
$878$	−7.18153	−0.242365
$879$	0	0
$880$	0.326222	0.0109970
$881$	28.9913	0.976740	0.488370	−	0.872637i	$-0.337592\pi$
$881$	28.9913	0.976740	0.488370	+	0.872637i	$0.337592\pi$
$882$	0	0
$883$	−17.1126	−0.575885	−0.287943	−	0.957648i	$-0.592971\pi$
$883$	−17.1126	−0.575885	−0.287943	+	0.957648i	$0.592971\pi$
$884$	−43.5194	−1.46372
$885$	0	0
$886$	−0.381336	−0.0128112
$887$	1.13794	0.0382082	0.0191041	−	0.999818i	$-0.493919\pi$
$887$	1.13794	0.0382082	0.0191041	+	0.999818i	$0.493919\pi$
$888$	0	0
$889$	9.96719	0.334289
$890$	−7.48924	−0.251040
$891$	0	0
$892$	−17.3461	−0.580791
$893$	1.00000	0.0334637
$894$	0	0
$895$	−15.5121	−0.518512
$896$	−5.57915	−0.186386
$897$	0	0
$898$	−3.03971	−0.101436
$899$	11.0470	0.368439
$900$	0	0
$901$	−30.2095	−1.00642
$902$	0.0273330	0.000910091 0
$903$	0	0
$904$	7.14709	0.237709
$905$	−49.8250	−1.65624
$906$	0	0
$907$	−13.9164	−0.462088	−0.231044	−	0.972943i	$-0.574214\pi$
$907$	−13.9164	−0.462088	−0.231044	+	0.972943i	$0.574214\pi$
$908$	41.4186	1.37452
$909$	0	0
$910$	−2.58036	−0.0855379
$911$	−0.385851	−0.0127838	−0.00639191	−	0.999980i	$-0.502035\pi$
$911$	−0.385851	−0.0127838	−0.00639191	+	0.999980i	$0.502035\pi$
$912$	0	0
$913$	−0.346513	−0.0114679
$914$	−0.706714	−0.0233760
$915$	0	0
$916$	−10.6317	−0.351281
$917$	−6.55163	−0.216354
$918$	0	0
$919$	−13.5451	−0.446812	−0.223406	−	0.974725i	$-0.571718\pi$
$919$	−13.5451	−0.446812	−0.223406	+	0.974725i	$0.571718\pi$
$920$	5.65516	0.186445
$921$	0	0
$922$	−1.39397	−0.0459080
$923$	−16.0789	−0.529243
$924$	0	0
$925$	8.60948	0.283078
$926$	6.60043	0.216904
$927$	0	0
$928$	12.8649	0.422310
$929$	54.1211	1.77565	0.887827	−	0.460177i	$-0.152214\pi$
$929$	54.1211	1.77565	0.887827	+	0.460177i	$0.152214\pi$
$930$	0	0
$931$	6.34268	0.207873
$932$	25.8023	0.845181
$933$	0	0
$934$	−3.14297	−0.102841
$935$	−0.356252	−0.0116507
$936$	0	0
$937$	−13.2315	−0.432253	−0.216126	−	0.976365i	$-0.569342\pi$
$937$	−13.2315	−0.432253	−0.216126	+	0.976365i	$0.569342\pi$
$938$	1.47520	0.0481669
$939$	0	0
$940$	4.85114	0.158227
$941$	13.3351	0.434710	0.217355	−	0.976093i	$-0.430257\pi$
$941$	13.3351	0.434710	0.217355	+	0.976093i	$0.430257\pi$
$942$	0	0
$943$	−8.36176	−0.272296
$944$	−2.81951	−0.0917672
$945$	0	0
$946$	0.0196034	0.000637362 0
$947$	42.6631	1.38636	0.693182	−	0.720762i	$-0.256208\pi$
$947$	42.6631	1.38636	0.693182	+	0.720762i	$0.256208\pi$
$948$	0	0
$949$	−75.7234	−2.45808
$950$	0.277740	0.00901108
$951$	0	0
$952$	2.96243	0.0960130
$953$	−42.2658	−1.36912	−0.684561	−	0.728955i	$-0.740006\pi$
$953$	−42.2658	−1.36912	−0.684561	+	0.728955i	$0.740006\pi$
$954$	0	0
$955$	33.3559	1.07937
$956$	2.08420	0.0674077
$957$	0	0
$958$	−4.36636	−0.141071
$959$	−6.12938	−0.197928
$960$	0	0
$961$	−25.7690	−0.831258
$962$	9.10824	0.293662
$963$	0	0
$964$	−34.2161	−1.10203
$965$	30.1792	0.971503
$966$	0	0
$967$	5.80004	0.186517	0.0932584	−	0.995642i	$-0.470272\pi$
$967$	5.80004	0.186517	0.0932584	+	0.995642i	$0.470272\pi$
$968$	−9.98577	−0.320955
$969$	0	0
$970$	0.223312	0.00717010
$971$	−3.39588	−0.108979	−0.0544896	−	0.998514i	$-0.517353\pi$
$971$	−3.39588	−0.108979	−0.0544896	+	0.998514i	$0.517353\pi$
$972$	0	0
$973$	−16.6086	−0.532446
$974$	−7.34090	−0.235217
$975$	0	0
$976$	33.8170	1.08245
$977$	−40.2847	−1.28882	−0.644410	−	0.764680i	$-0.722897\pi$
$977$	−40.2847	−1.28882	−0.644410	+	0.764680i	$0.722897\pi$
$978$	0	0
$979$	0.464298	0.0148390
$980$	30.7692	0.982888
$981$	0	0
$982$	−0.0910895	−0.00290678
$983$	23.5934	0.752513	0.376257	−	0.926516i	$-0.377211\pi$
$983$	23.5934	0.752513	0.376257	+	0.926516i	$0.377211\pi$
$984$	0	0
$985$	37.0932	1.18189
$986$	−4.47133	−0.142396
$987$	0	0
$988$	−10.8133	−0.344018
$989$	−5.99711	−0.190697
$990$	0	0
$991$	−17.4357	−0.553862	−0.276931	−	0.960890i	$-0.589317\pi$
$991$	−17.4357	−0.553862	−0.276931	+	0.960890i	$0.589317\pi$
$992$	6.09180	0.193415
$993$	0	0
$994$	0.539923	0.0171253
$995$	14.4425	0.457859
$996$	0	0
$997$	24.8425	0.786769	0.393384	−	0.919374i	$-0.371304\pi$
$997$	24.8425	0.786769	0.393384	+	0.919374i	$0.371304\pi$
$998$	8.30445	0.262873
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.o.1.8		18
3.2	odd	2		893.2.a.c.1.11	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.c.1.11	✓	18	3.2	odd	2
8037.2.a.o.1.8		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)

\(f(q)\)	\(=\)	\(q-0.230018 q^{2} -1.94709 q^{4} +2.49148 q^{5} -0.810750 q^{7} +0.907901 q^{8} +O(q^{10})\)	\(q-0.230018 q^{2} -1.94709 q^{4} +2.49148 q^{5} -0.810750 q^{7} +0.907901 q^{8} -0.573085 q^{10} +0.0355286 q^{11} -5.55359 q^{13} +0.186487 q^{14} +3.68535 q^{16} -4.02460 q^{17} -1.00000 q^{19} -4.85114 q^{20} -0.00817220 q^{22} +2.50005 q^{23} +1.20747 q^{25} +1.27742 q^{26} +1.57861 q^{28} -4.83006 q^{29} -2.28714 q^{31} -2.66350 q^{32} +0.925730 q^{34} -2.01997 q^{35} +7.13017 q^{37} +0.230018 q^{38} +2.26202 q^{40} -3.34464 q^{41} -2.39879 q^{43} -0.0691774 q^{44} -0.575056 q^{46} -1.00000 q^{47} -6.34268 q^{49} -0.277740 q^{50} +10.8133 q^{52} +7.50620 q^{53} +0.0885187 q^{55} -0.736081 q^{56} +1.11100 q^{58} -0.765058 q^{59} +9.17605 q^{61} +0.526083 q^{62} -6.75805 q^{64} -13.8366 q^{65} +7.91045 q^{67} +7.83627 q^{68} +0.464629 q^{70} +2.89523 q^{71} +13.6350 q^{73} -1.64007 q^{74} +1.94709 q^{76} -0.0288048 q^{77} -3.12962 q^{79} +9.18198 q^{80} +0.769326 q^{82} -9.75309 q^{83} -10.0272 q^{85} +0.551765 q^{86} +0.0322564 q^{88} +13.0683 q^{89} +4.50257 q^{91} -4.86783 q^{92} +0.230018 q^{94} -2.49148 q^{95} -0.389666 q^{97} +1.45893 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 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

Related objects

Downloads

Learn more