LMFDB - Embedded newform 8037.2.a.o.1.12

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			1.12
Root			$-1.15272$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.15272 q^{2} -0.671235 q^{4} +1.54521 q^{5} +2.91562 q^{7} -3.07919 q^{8} +O(q^{10})$	\(q+1.15272 q^{2} -0.671235 q^{4} +1.54521 q^{5} +2.91562 q^{7} -3.07919 q^{8} +1.78119 q^{10} +3.95117 q^{11} +6.00224 q^{13} +3.36090 q^{14} -2.20697 q^{16} -1.75048 q^{17} -1.00000 q^{19} -1.03720 q^{20} +4.55460 q^{22} -4.20228 q^{23} -2.61234 q^{25} +6.91890 q^{26} -1.95706 q^{28} +5.44146 q^{29} +7.26625 q^{31} +3.61435 q^{32} -2.01782 q^{34} +4.50523 q^{35} +3.75073 q^{37} -1.15272 q^{38} -4.75798 q^{40} -10.9655 q^{41} +11.9908 q^{43} -2.65216 q^{44} -4.84405 q^{46} -1.00000 q^{47} +1.50084 q^{49} -3.01129 q^{50} -4.02891 q^{52} +2.20312 q^{53} +6.10538 q^{55} -8.97774 q^{56} +6.27248 q^{58} +1.96455 q^{59} +12.7938 q^{61} +8.37596 q^{62} +8.58029 q^{64} +9.27470 q^{65} -4.42593 q^{67} +1.17498 q^{68} +5.19328 q^{70} -6.41036 q^{71} +0.148533 q^{73} +4.32354 q^{74} +0.671235 q^{76} +11.5201 q^{77} -5.77923 q^{79} -3.41023 q^{80} -12.6401 q^{82} +3.42533 q^{83} -2.70486 q^{85} +13.8221 q^{86} -12.1664 q^{88} +10.1974 q^{89} +17.5002 q^{91} +2.82071 q^{92} -1.15272 q^{94} -1.54521 q^{95} -3.24912 q^{97} +1.73005 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.15272	0.815097	0.407548	−	0.913184i	$-0.366384\pi$
$2$	1.15272	0.815097	0.407548	+	0.913184i	$0.366384\pi$
$3$	0	0
$3$	0	0
$4$	−0.671235	−0.335617
$5$	1.54521	0.691037	0.345519	−	0.938412i	$-0.387703\pi$
$5$	1.54521	0.691037	0.345519	+	0.938412i	$0.387703\pi$
$6$	0	0
$7$	2.91562	1.10200	0.551000	−	0.834505i	$-0.314246\pi$
$7$	2.91562	1.10200	0.551000	+	0.834505i	$0.314246\pi$
$8$	−3.07919	−1.08866
$9$	0	0
$10$	1.78119	0.563262
$11$	3.95117	1.19132	0.595662	−	0.803236i	$-0.296890\pi$
$11$	3.95117	1.19132	0.595662	+	0.803236i	$0.296890\pi$
$12$	0	0
$13$	6.00224	1.66472	0.832361	−	0.554234i	$-0.186989\pi$
$13$	6.00224	1.66472	0.832361	+	0.554234i	$0.186989\pi$
$14$	3.36090	0.898237
$15$	0	0
$16$	−2.20697	−0.551744
$17$	−1.75048	−0.424554	−0.212277	−	0.977210i	$-0.568088\pi$
$17$	−1.75048	−0.424554	−0.212277	+	0.977210i	$0.568088\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.03720	−0.231924
$21$	0	0
$22$	4.55460	0.971044
$23$	−4.20228	−0.876235	−0.438117	−	0.898918i	$-0.644355\pi$
$23$	−4.20228	−0.876235	−0.438117	+	0.898918i	$0.644355\pi$
$24$	0	0
$25$	−2.61234	−0.522467
$26$	6.91890	1.35691
$27$	0	0
$28$	−1.95706	−0.369851
$29$	5.44146	1.01045	0.505227	−	0.862987i	$-0.331409\pi$
$29$	5.44146	1.01045	0.505227	+	0.862987i	$0.331409\pi$
$30$	0	0
$31$	7.26625	1.30506	0.652528	−	0.757764i	$-0.273708\pi$
$31$	7.26625	1.30506	0.652528	+	0.757764i	$0.273708\pi$
$32$	3.61435	0.638933
$33$	0	0
$34$	−2.01782	−0.346053
$35$	4.50523	0.761524
$36$	0	0
$37$	3.75073	0.616615	0.308308	−	0.951287i	$-0.400237\pi$
$37$	3.75073	0.616615	0.308308	+	0.951287i	$0.400237\pi$
$38$	−1.15272	−0.186996
$39$	0	0
$40$	−4.75798	−0.752303
$41$	−10.9655	−1.71252	−0.856260	−	0.516546i	$-0.827218\pi$
$41$	−10.9655	−1.71252	−0.856260	+	0.516546i	$0.827218\pi$
$42$	0	0
$43$	11.9908	1.82859	0.914293	−	0.405053i	$-0.132747\pi$
$43$	11.9908	1.82859	0.914293	+	0.405053i	$0.132747\pi$
$44$	−2.65216	−0.399829
$45$	0	0
$46$	−4.84405	−0.714216
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	1.50084	0.214405
$50$	−3.01129	−0.425861
$51$	0	0
$52$	−4.02891	−0.558709
$53$	2.20312	0.302621	0.151311	−	0.988486i	$-0.451651\pi$
$53$	2.20312	0.302621	0.151311	+	0.988486i	$0.451651\pi$
$54$	0	0
$55$	6.10538	0.823249
$56$	−8.97774	−1.19970
$57$	0	0
$58$	6.27248	0.823617
$59$	1.96455	0.255762	0.127881	−	0.991790i	$-0.459182\pi$
$59$	1.96455	0.255762	0.127881	+	0.991790i	$0.459182\pi$
$60$	0	0
$61$	12.7938	1.63808	0.819038	−	0.573740i	$-0.194508\pi$
$61$	12.7938	1.63808	0.819038	+	0.573740i	$0.194508\pi$
$62$	8.37596	1.06375
$63$	0	0
$64$	8.58029	1.07254
$65$	9.27470	1.15038
$66$	0	0
$67$	−4.42593	−0.540713	−0.270357	−	0.962760i	$-0.587142\pi$
$67$	−4.42593	−0.540713	−0.270357	+	0.962760i	$0.587142\pi$
$68$	1.17498	0.142488
$69$	0	0
$70$	5.19328	0.620715
$71$	−6.41036	−0.760770	−0.380385	−	0.924828i	$-0.624209\pi$
$71$	−6.41036	−0.760770	−0.380385	+	0.924828i	$0.624209\pi$
$72$	0	0
$73$	0.148533	0.0173844	0.00869222	−	0.999962i	$-0.497233\pi$
$73$	0.148533	0.0173844	0.00869222	+	0.999962i	$0.497233\pi$
$74$	4.32354	0.502601
$75$	0	0
$76$	0.671235	0.0769959
$77$	11.5201	1.31284
$78$	0	0
$79$	−5.77923	−0.650214	−0.325107	−	0.945677i	$-0.605400\pi$
$79$	−5.77923	−0.650214	−0.325107	+	0.945677i	$0.605400\pi$
$80$	−3.41023	−0.381276
$81$	0	0
$82$	−12.6401	−1.39587
$83$	3.42533	0.375979	0.187990	−	0.982171i	$-0.439803\pi$
$83$	3.42533	0.375979	0.187990	+	0.982171i	$0.439803\pi$
$84$	0	0
$85$	−2.70486	−0.293383
$86$	13.8221	1.49047
$87$	0	0
$88$	−12.1664	−1.29694
$89$	10.1974	1.08092	0.540459	−	0.841370i	$-0.318250\pi$
$89$	10.1974	1.08092	0.540459	+	0.841370i	$0.318250\pi$
$90$	0	0
$91$	17.5002	1.83452
$92$	2.82071	0.294080
$93$	0	0
$94$	−1.15272	−0.118894
$95$	−1.54521	−0.158535
$96$	0	0
$97$	−3.24912	−0.329899	−0.164949	−	0.986302i	$-0.552746\pi$
$97$	−3.24912	−0.329899	−0.164949	+	0.986302i	$0.552746\pi$
$98$	1.73005	0.174761
$99$	0	0
$100$	1.75349	0.175349
$101$	−10.1357	−1.00854	−0.504268	−	0.863547i	$-0.668238\pi$
$101$	−10.1357	−1.00854	−0.504268	+	0.863547i	$0.668238\pi$
$102$	0	0
$103$	−7.50903	−0.739887	−0.369943	−	0.929054i	$-0.620623\pi$
$103$	−7.50903	−0.739887	−0.369943	+	0.929054i	$0.620623\pi$
$104$	−18.4820	−1.81231
$105$	0	0
$106$	2.53958	0.246666
$107$	−19.9631	−1.92991	−0.964953	−	0.262424i	$-0.915478\pi$
$107$	−19.9631	−1.92991	−0.964953	+	0.262424i	$0.915478\pi$
$108$	0	0
$109$	−4.35081	−0.416732	−0.208366	−	0.978051i	$-0.566814\pi$
$109$	−4.35081	−0.416732	−0.208366	+	0.978051i	$0.566814\pi$
$110$	7.03780	0.671027
$111$	0	0
$112$	−6.43470	−0.608022
$113$	6.91095	0.650128	0.325064	−	0.945692i	$-0.394614\pi$
$113$	6.91095	0.650128	0.325064	+	0.945692i	$0.394614\pi$
$114$	0	0
$115$	−6.49338	−0.605511
$116$	−3.65250	−0.339126
$117$	0	0
$118$	2.26457	0.208471
$119$	−5.10374	−0.467859
$120$	0	0
$121$	4.61176	0.419251
$122$	14.7477	1.33519
$123$	0	0
$124$	−4.87736	−0.438000
$125$	−11.7626	−1.05208
$126$	0	0
$127$	−4.98894	−0.442697	−0.221349	−	0.975195i	$-0.571046\pi$
$127$	−4.98894	−0.442697	−0.221349	+	0.975195i	$0.571046\pi$
$128$	2.66197	0.235287
$129$	0	0
$130$	10.6911	0.937675
$131$	6.06719	0.530093	0.265046	−	0.964236i	$-0.414613\pi$
$131$	6.06719	0.530093	0.265046	+	0.964236i	$0.414613\pi$
$132$	0	0
$133$	−2.91562	−0.252816
$134$	−5.10186	−0.440733
$135$	0	0
$136$	5.39006	0.462194
$137$	16.5379	1.41293	0.706465	−	0.707748i	$-0.250289\pi$
$137$	16.5379	1.41293	0.706465	+	0.707748i	$0.250289\pi$
$138$	0	0
$139$	11.2019	0.950129	0.475064	−	0.879951i	$-0.342425\pi$
$139$	11.2019	0.950129	0.475064	+	0.879951i	$0.342425\pi$
$140$	−3.02407	−0.255581
$141$	0	0
$142$	−7.38936	−0.620101
$143$	23.7159	1.98322
$144$	0	0
$145$	8.40818	0.698261
$146$	0.171217	0.0141700
$147$	0	0
$148$	−2.51762	−0.206947
$149$	−10.5839	−0.867070	−0.433535	−	0.901137i	$-0.642734\pi$
$149$	−10.5839	−0.867070	−0.433535	+	0.901137i	$0.642734\pi$
$150$	0	0
$151$	−2.05452	−0.167195	−0.0835974	−	0.996500i	$-0.526641\pi$
$151$	−2.05452	−0.167195	−0.0835974	+	0.996500i	$0.526641\pi$
$152$	3.07919	0.249755
$153$	0	0
$154$	13.2795	1.07009
$155$	11.2279	0.901843
$156$	0	0
$157$	−12.6222	−1.00736	−0.503679	−	0.863891i	$-0.668020\pi$
$157$	−12.6222	−1.00736	−0.503679	+	0.863891i	$0.668020\pi$
$158$	−6.66184	−0.529987
$159$	0	0
$160$	5.58492	0.441527
$161$	−12.2522	−0.965611
$162$	0	0
$163$	−10.2629	−0.803852	−0.401926	−	0.915672i	$-0.631659\pi$
$163$	−10.2629	−0.803852	−0.401926	+	0.915672i	$0.631659\pi$
$164$	7.36041	0.574751
$165$	0	0
$166$	3.94845	0.306459
$167$	22.2308	1.72027	0.860134	−	0.510069i	$-0.170380\pi$
$167$	22.2308	1.72027	0.860134	+	0.510069i	$0.170380\pi$
$168$	0	0
$169$	23.0269	1.77130
$170$	−3.11794	−0.239135
$171$	0	0
$172$	−8.04867	−0.613705
$173$	1.38919	0.105618	0.0528092	−	0.998605i	$-0.483182\pi$
$173$	1.38919	0.105618	0.0528092	+	0.998605i	$0.483182\pi$
$174$	0	0
$175$	−7.61658	−0.575759
$176$	−8.72014	−0.657305
$177$	0	0
$178$	11.7547	0.881053
$179$	8.78035	0.656274	0.328137	−	0.944630i	$-0.393579\pi$
$179$	8.78035	0.656274	0.328137	+	0.944630i	$0.393579\pi$
$180$	0	0
$181$	−17.8722	−1.32843	−0.664216	−	0.747540i	$-0.731235\pi$
$181$	−17.8722	−1.32843	−0.664216	+	0.747540i	$0.731235\pi$
$182$	20.1729	1.49531
$183$	0	0
$184$	12.9396	0.953920
$185$	5.79565	0.426104
$186$	0	0
$187$	−6.91646	−0.505781
$188$	0.671235	0.0489548
$189$	0	0
$190$	−1.78119	−0.129221
$191$	10.7090	0.774879	0.387439	−	0.921895i	$-0.373360\pi$
$191$	10.7090	0.774879	0.387439	+	0.921895i	$0.373360\pi$
$192$	0	0
$193$	19.1513	1.37854	0.689271	−	0.724504i	$-0.257931\pi$
$193$	19.1513	1.37854	0.689271	+	0.724504i	$0.257931\pi$
$194$	−3.74533	−0.268899
$195$	0	0
$196$	−1.00741	−0.0719581
$197$	−4.43638	−0.316079	−0.158039	−	0.987433i	$-0.550517\pi$
$197$	−4.43638	−0.316079	−0.158039	+	0.987433i	$0.550517\pi$
$198$	0	0
$199$	−26.6710	−1.89066	−0.945329	−	0.326117i	$-0.894260\pi$
$199$	−26.6710	−1.89066	−0.945329	+	0.326117i	$0.894260\pi$
$200$	8.04387	0.568788
$201$	0	0
$202$	−11.6836	−0.822054
$203$	15.8652	1.11352
$204$	0	0
$205$	−16.9439	−1.18342
$206$	−8.65582	−0.603079
$207$	0	0
$208$	−13.2468	−0.918499
$209$	−3.95117	−0.273308
$210$	0	0
$211$	21.8926	1.50715	0.753573	−	0.657365i	$-0.228329\pi$
$211$	21.8926	1.50715	0.753573	+	0.657365i	$0.228329\pi$
$212$	−1.47881	−0.101565
$213$	0	0
$214$	−23.0119	−1.57306
$215$	18.5283	1.26362
$216$	0	0
$217$	21.1856	1.43817
$218$	−5.01526	−0.339677
$219$	0	0
$220$	−4.09814	−0.276297
$221$	−10.5068	−0.706764
$222$	0	0
$223$	−3.86818	−0.259033	−0.129516	−	0.991577i	$-0.541342\pi$
$223$	−3.86818	−0.259033	−0.129516	+	0.991577i	$0.541342\pi$
$224$	10.5381	0.704104
$225$	0	0
$226$	7.96640	0.529917
$227$	6.36395	0.422390	0.211195	−	0.977444i	$-0.432264\pi$
$227$	6.36395	0.422390	0.211195	+	0.977444i	$0.432264\pi$
$228$	0	0
$229$	12.8601	0.849818	0.424909	−	0.905236i	$-0.360306\pi$
$229$	12.8601	0.849818	0.424909	+	0.905236i	$0.360306\pi$
$230$	−7.48506	−0.493550
$231$	0	0
$232$	−16.7553	−1.10004
$233$	−22.3801	−1.46617	−0.733085	−	0.680137i	$-0.761920\pi$
$233$	−22.3801	−1.46617	−0.733085	+	0.680137i	$0.761920\pi$
$234$	0	0
$235$	−1.54521	−0.100798
$236$	−1.31867	−0.0858382
$237$	0	0
$238$	−5.88319	−0.381350
$239$	19.3132	1.24927	0.624634	−	0.780918i	$-0.285248\pi$
$239$	19.3132	1.24927	0.624634	+	0.780918i	$0.285248\pi$
$240$	0	0
$241$	−15.0525	−0.969619	−0.484809	−	0.874620i	$-0.661111\pi$
$241$	−15.0525	−0.969619	−0.484809	+	0.874620i	$0.661111\pi$
$242$	5.31607	0.341730
$243$	0	0
$244$	−8.58763	−0.549767
$245$	2.31910	0.148162
$246$	0	0
$247$	−6.00224	−0.381913
$248$	−22.3741	−1.42076
$249$	0	0
$250$	−13.5590	−0.857548
$251$	12.4706	0.787138	0.393569	−	0.919295i	$-0.371240\pi$
$251$	12.4706	0.787138	0.393569	+	0.919295i	$0.371240\pi$
$252$	0	0
$253$	−16.6039	−1.04388
$254$	−5.75086	−0.360841
$255$	0	0
$256$	−14.0921	−0.880754
$257$	13.1686	0.821433	0.410716	−	0.911763i	$-0.365279\pi$
$257$	13.1686	0.821433	0.410716	+	0.911763i	$0.365279\pi$
$258$	0	0
$259$	10.9357	0.679511
$260$	−6.22550	−0.386089
$261$	0	0
$262$	6.99378	0.432077
$263$	9.57085	0.590164	0.295082	−	0.955472i	$-0.404653\pi$
$263$	9.57085	0.590164	0.295082	+	0.955472i	$0.404653\pi$
$264$	0	0
$265$	3.40427	0.209123
$266$	−3.36090	−0.206070
$267$	0	0
$268$	2.97084	0.181473
$269$	8.89380	0.542265	0.271132	−	0.962542i	$-0.412602\pi$
$269$	8.89380	0.542265	0.271132	+	0.962542i	$0.412602\pi$
$270$	0	0
$271$	−0.952065	−0.0578338	−0.0289169	−	0.999582i	$-0.509206\pi$
$271$	−0.952065	−0.0578338	−0.0289169	+	0.999582i	$0.509206\pi$
$272$	3.86327	0.234245
$273$	0	0
$274$	19.0636	1.15167
$275$	−10.3218	−0.622427
$276$	0	0
$277$	13.1513	0.790186	0.395093	−	0.918641i	$-0.370712\pi$
$277$	13.1513	0.790186	0.395093	+	0.918641i	$0.370712\pi$
$278$	12.9126	0.774447
$279$	0	0
$280$	−13.8725	−0.829038
$281$	25.7448	1.53581	0.767903	−	0.640566i	$-0.221300\pi$
$281$	25.7448	1.53581	0.767903	+	0.640566i	$0.221300\pi$
$282$	0	0
$283$	−11.7047	−0.695772	−0.347886	−	0.937537i	$-0.613100\pi$
$283$	−11.7047	−0.695772	−0.347886	+	0.937537i	$0.613100\pi$
$284$	4.30286	0.255328
$285$	0	0
$286$	27.3378	1.61652
$287$	−31.9712	−1.88720
$288$	0	0
$289$	−13.9358	−0.819754
$290$	9.69228	0.569150
$291$	0	0
$292$	−0.0997004	−0.00583452
$293$	19.0990	1.11578	0.557888	−	0.829917i	$-0.311612\pi$
$293$	19.0990	1.11578	0.557888	+	0.829917i	$0.311612\pi$
$294$	0	0
$295$	3.03563	0.176741
$296$	−11.5492	−0.671283
$297$	0	0
$298$	−12.2003	−0.706746
$299$	−25.2231	−1.45869
$300$	0	0
$301$	34.9607	2.01510
$302$	−2.36829	−0.136280
$303$	0	0
$304$	2.20697	0.126579
$305$	19.7690	1.13197
$306$	0	0
$307$	5.51798	0.314928	0.157464	−	0.987525i	$-0.449668\pi$
$307$	5.51798	0.314928	0.157464	+	0.987525i	$0.449668\pi$
$308$	−7.73270	−0.440611
$309$	0	0
$310$	12.9426	0.735089
$311$	−3.05334	−0.173139	−0.0865696	−	0.996246i	$-0.527590\pi$
$311$	−3.05334	−0.173139	−0.0865696	+	0.996246i	$0.527590\pi$
$312$	0	0
$313$	31.2294	1.76519	0.882595	−	0.470134i	$-0.155795\pi$
$313$	31.2294	1.76519	0.882595	+	0.470134i	$0.155795\pi$
$314$	−14.5498	−0.821094
$315$	0	0
$316$	3.87922	0.218223
$317$	18.5408	1.04135	0.520676	−	0.853754i	$-0.325680\pi$
$317$	18.5408	1.04135	0.520676	+	0.853754i	$0.325680\pi$
$318$	0	0
$319$	21.5001	1.20378
$320$	13.2583	0.741162
$321$	0	0
$322$	−14.1234	−0.787067
$323$	1.75048	0.0973994
$324$	0	0
$325$	−15.6799	−0.869762
$326$	−11.8303	−0.655217
$327$	0	0
$328$	33.7648	1.86435
$329$	−2.91562	−0.160743
$330$	0	0
$331$	−8.94549	−0.491688	−0.245844	−	0.969309i	$-0.579065\pi$
$331$	−8.94549	−0.491688	−0.245844	+	0.969309i	$0.579065\pi$
$332$	−2.29920	−0.126185
$333$	0	0
$334$	25.6259	1.40218
$335$	−6.83897	−0.373653
$336$	0	0
$337$	−14.0355	−0.764562	−0.382281	−	0.924046i	$-0.624861\pi$
$337$	−14.0355	−0.764562	−0.382281	+	0.924046i	$0.624861\pi$
$338$	26.5435	1.44378
$339$	0	0
$340$	1.81559	0.0984644
$341$	28.7102	1.55474
$342$	0	0
$343$	−16.0335	−0.865726
$344$	−36.9221	−1.99070
$345$	0	0
$346$	1.60135	0.0860892
$347$	−25.5836	−1.37340	−0.686699	−	0.726942i	$-0.740941\pi$
$347$	−25.5836	−1.37340	−0.686699	+	0.726942i	$0.740941\pi$
$348$	0	0
$349$	22.6907	1.21460	0.607302	−	0.794471i	$-0.292252\pi$
$349$	22.6907	1.21460	0.607302	+	0.794471i	$0.292252\pi$
$350$	−8.77979	−0.469299
$351$	0	0
$352$	14.2809	0.761175
$353$	34.2603	1.82349	0.911746	−	0.410754i	$-0.134734\pi$
$353$	34.2603	1.82349	0.911746	+	0.410754i	$0.134734\pi$
$354$	0	0
$355$	−9.90534	−0.525721
$356$	−6.84483	−0.362775
$357$	0	0
$358$	10.1213	0.534927
$359$	31.4986	1.66243	0.831216	−	0.555950i	$-0.187645\pi$
$359$	31.4986	1.66243	0.831216	+	0.555950i	$0.187645\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−20.6017	−1.08280
$363$	0	0
$364$	−11.7468	−0.615698
$365$	0.229514	0.0120133
$366$	0	0
$367$	19.1134	0.997714	0.498857	−	0.866684i	$-0.333753\pi$
$367$	19.1134	0.997714	0.498857	+	0.866684i	$0.333753\pi$
$368$	9.27431	0.483457
$369$	0	0
$370$	6.68076	0.347316
$371$	6.42345	0.333489
$372$	0	0
$373$	15.3890	0.796815	0.398407	−	0.917209i	$-0.369563\pi$
$373$	15.3890	0.796815	0.398407	+	0.917209i	$0.369563\pi$
$374$	−7.97274	−0.412261
$375$	0	0
$376$	3.07919	0.158797
$377$	32.6609	1.68212
$378$	0	0
$379$	0.412947	0.0212117	0.0106058	−	0.999944i	$-0.496624\pi$
$379$	0.412947	0.0212117	0.0106058	+	0.999944i	$0.496624\pi$
$380$	1.03720	0.0532071
$381$	0	0
$382$	12.3445	0.631601
$383$	1.91879	0.0980454	0.0490227	−	0.998798i	$-0.484389\pi$
$383$	1.91879	0.0980454	0.0490227	+	0.998798i	$0.484389\pi$
$384$	0	0
$385$	17.8010	0.907221
$386$	22.0761	1.12364
$387$	0	0
$388$	2.18092	0.110720
$389$	5.59847	0.283854	0.141927	−	0.989877i	$-0.454670\pi$
$389$	5.59847	0.283854	0.141927	+	0.989877i	$0.454670\pi$
$390$	0	0
$391$	7.35601	0.372009
$392$	−4.62136	−0.233414
$393$	0	0
$394$	−5.11391	−0.257635
$395$	−8.93010	−0.449322
$396$	0	0
$397$	−27.2497	−1.36762	−0.683811	−	0.729659i	$-0.739679\pi$
$397$	−27.2497	−1.36762	−0.683811	+	0.729659i	$0.739679\pi$
$398$	−30.7442	−1.54107
$399$	0	0
$400$	5.76536	0.288268
$401$	−24.7824	−1.23757	−0.618787	−	0.785559i	$-0.712376\pi$
$401$	−24.7824	−1.23757	−0.618787	+	0.785559i	$0.712376\pi$
$402$	0	0
$403$	43.6138	2.17256
$404$	6.80341	0.338482
$405$	0	0
$406$	18.2882	0.907627
$407$	14.8198	0.734588
$408$	0	0
$409$	24.3663	1.20484	0.602419	−	0.798180i	$-0.294204\pi$
$409$	24.3663	1.20484	0.602419	+	0.798180i	$0.294204\pi$
$410$	−19.5316	−0.964598
$411$	0	0
$412$	5.04032	0.248319
$413$	5.72787	0.281850
$414$	0	0
$415$	5.29285	0.259816
$416$	21.6942	1.06365
$417$	0	0
$418$	−4.55460	−0.222773
$419$	−37.6896	−1.84126	−0.920629	−	0.390439i	$-0.872323\pi$
$419$	−37.6896	−1.84126	−0.920629	+	0.390439i	$0.872323\pi$
$420$	0	0
$421$	27.3056	1.33079	0.665396	−	0.746490i	$-0.268263\pi$
$421$	27.3056	1.33079	0.665396	+	0.746490i	$0.268263\pi$
$422$	25.2360	1.22847
$423$	0	0
$424$	−6.78381	−0.329451
$425$	4.57285	0.221816
$426$	0	0
$427$	37.3018	1.80516
$428$	13.3999	0.647710
$429$	0	0
$430$	21.3580	1.02997
$431$	−23.6012	−1.13683	−0.568414	−	0.822743i	$-0.692443\pi$
$431$	−23.6012	−1.13683	−0.568414	+	0.822743i	$0.692443\pi$
$432$	0	0
$433$	−20.7137	−0.995437	−0.497718	−	0.867339i	$-0.665829\pi$
$433$	−20.7137	−0.995437	−0.497718	+	0.867339i	$0.665829\pi$
$434$	24.4211	1.17225
$435$	0	0
$436$	2.92041	0.139862
$437$	4.20228	0.201022
$438$	0	0
$439$	27.4529	1.31026	0.655129	−	0.755517i	$-0.272614\pi$
$439$	27.4529	1.31026	0.655129	+	0.755517i	$0.272614\pi$
$440$	−18.7996	−0.896236
$441$	0	0
$442$	−12.1114	−0.576081
$443$	−9.57671	−0.455003	−0.227502	−	0.973778i	$-0.573056\pi$
$443$	−9.57671	−0.455003	−0.227502	+	0.973778i	$0.573056\pi$
$444$	0	0
$445$	15.7570	0.746955
$446$	−4.45894	−0.211137
$447$	0	0
$448$	25.0168	1.18193
$449$	4.50798	0.212745	0.106372	−	0.994326i	$-0.466076\pi$
$449$	4.50798	0.212745	0.106372	+	0.994326i	$0.466076\pi$
$450$	0	0
$451$	−43.3265	−2.04016
$452$	−4.63887	−0.218194
$453$	0	0
$454$	7.33586	0.344289
$455$	27.0415	1.26772
$456$	0	0
$457$	−37.4339	−1.75108	−0.875541	−	0.483143i	$-0.839495\pi$
$457$	−37.4339	−1.75108	−0.875541	+	0.483143i	$0.839495\pi$
$458$	14.8241	0.692684
$459$	0	0
$460$	4.35858	0.203220
$461$	7.94938	0.370240	0.185120	−	0.982716i	$-0.440733\pi$
$461$	7.94938	0.370240	0.185120	+	0.982716i	$0.440733\pi$
$462$	0	0
$463$	1.70643	0.0793046	0.0396523	−	0.999214i	$-0.487375\pi$
$463$	1.70643	0.0793046	0.0396523	+	0.999214i	$0.487375\pi$
$464$	−12.0092	−0.557511
$465$	0	0
$466$	−25.7980	−1.19507
$467$	−24.8522	−1.15002	−0.575011	−	0.818146i	$-0.695002\pi$
$467$	−24.8522	−1.15002	−0.575011	+	0.818146i	$0.695002\pi$
$468$	0	0
$469$	−12.9043	−0.595866
$470$	−1.78119	−0.0821603
$471$	0	0
$472$	−6.04921	−0.278437
$473$	47.3779	2.17844
$474$	0	0
$475$	2.61234	0.119862
$476$	3.42581	0.157022
$477$	0	0
$478$	22.2627	1.01827
$479$	23.4230	1.07023	0.535113	−	0.844781i	$-0.320269\pi$
$479$	23.4230	1.07023	0.535113	+	0.844781i	$0.320269\pi$
$480$	0	0
$481$	22.5127	1.02649
$482$	−17.3514	−0.790333
$483$	0	0
$484$	−3.09557	−0.140708
$485$	−5.02057	−0.227972
$486$	0	0
$487$	10.8497	0.491647	0.245823	−	0.969315i	$-0.420942\pi$
$487$	10.8497	0.491647	0.245823	+	0.969315i	$0.420942\pi$
$488$	−39.3944	−1.78330
$489$	0	0
$490$	2.67328	0.120766
$491$	−21.6561	−0.977327	−0.488664	−	0.872472i	$-0.662515\pi$
$491$	−21.6561	−0.977327	−0.488664	+	0.872472i	$0.662515\pi$
$492$	0	0
$493$	−9.52518	−0.428992
$494$	−6.91890	−0.311296
$495$	0	0
$496$	−16.0364	−0.720057
$497$	−18.6902	−0.838369
$498$	0	0
$499$	29.1821	1.30637	0.653185	−	0.757199i	$-0.273433\pi$
$499$	29.1821	1.30637	0.653185	+	0.757199i	$0.273433\pi$
$500$	7.89549	0.353097
$501$	0	0
$502$	14.3751	0.641593
$503$	12.3230	0.549454	0.274727	−	0.961522i	$-0.411413\pi$
$503$	12.3230	0.549454	0.274727	+	0.961522i	$0.411413\pi$
$504$	0	0
$505$	−15.6617	−0.696936
$506$	−19.1397	−0.850862
$507$	0	0
$508$	3.34875	0.148577
$509$	−24.5191	−1.08679	−0.543395	−	0.839477i	$-0.682861\pi$
$509$	−24.5191	−1.08679	−0.543395	+	0.839477i	$0.682861\pi$
$510$	0	0
$511$	0.433065	0.0191577
$512$	−21.5682	−0.953187
$513$	0	0
$514$	15.1797	0.669547
$515$	−11.6030	−0.511290
$516$	0	0
$517$	−3.95117	−0.173772
$518$	12.6058	0.553867
$519$	0	0
$520$	−28.5585	−1.25237
$521$	−37.1691	−1.62841	−0.814205	−	0.580578i	$-0.802827\pi$
$521$	−37.1691	−1.62841	−0.814205	+	0.580578i	$0.802827\pi$
$522$	0	0
$523$	−7.92415	−0.346499	−0.173249	−	0.984878i	$-0.555427\pi$
$523$	−7.92415	−0.346499	−0.173249	+	0.984878i	$0.555427\pi$
$524$	−4.07251	−0.177908
$525$	0	0
$526$	11.0325	0.481041
$527$	−12.7194	−0.554067
$528$	0	0
$529$	−5.34088	−0.232212
$530$	3.92417	0.170455
$531$	0	0
$532$	1.95706	0.0848495
$533$	−65.8174	−2.85087
$534$	0	0
$535$	−30.8471	−1.33364
$536$	13.6283	0.588651
$537$	0	0
$538$	10.2521	0.441998
$539$	5.93006	0.255426
$540$	0	0
$541$	−19.4796	−0.837491	−0.418746	−	0.908104i	$-0.637530\pi$
$541$	−19.4796	−0.837491	−0.418746	+	0.908104i	$0.637530\pi$
$542$	−1.09746	−0.0471402
$543$	0	0
$544$	−6.32686	−0.271262
$545$	−6.72289	−0.287977
$546$	0	0
$547$	−10.0105	−0.428020	−0.214010	−	0.976832i	$-0.568652\pi$
$547$	−10.0105	−0.428020	−0.214010	+	0.976832i	$0.568652\pi$
$548$	−11.1008	−0.474204
$549$	0	0
$550$	−11.8981	−0.507338
$551$	−5.44146	−0.231814
$552$	0	0
$553$	−16.8500	−0.716536
$554$	15.1598	0.644078
$555$	0	0
$556$	−7.51907	−0.318880
$557$	−2.92491	−0.123932	−0.0619662	−	0.998078i	$-0.519737\pi$
$557$	−2.92491	−0.123932	−0.0619662	+	0.998078i	$0.519737\pi$
$558$	0	0
$559$	71.9719	3.04409
$560$	−9.94294	−0.420166
$561$	0	0
$562$	29.6766	1.25183
$563$	−18.3400	−0.772938	−0.386469	−	0.922302i	$-0.626305\pi$
$563$	−18.3400	−0.772938	−0.386469	+	0.922302i	$0.626305\pi$
$564$	0	0
$565$	10.6788	0.449263
$566$	−13.4923	−0.567122
$567$	0	0
$568$	19.7387	0.828218
$569$	−17.7828	−0.745495	−0.372748	−	0.927933i	$-0.621584\pi$
$569$	−17.7828	−0.745495	−0.372748	+	0.927933i	$0.621584\pi$
$570$	0	0
$571$	−32.0472	−1.34113	−0.670567	−	0.741849i	$-0.733949\pi$
$571$	−32.0472	−1.34113	−0.670567	+	0.741849i	$0.733949\pi$
$572$	−15.9189	−0.665603
$573$	0	0
$574$	−36.8538	−1.53825
$575$	10.9778	0.457804
$576$	0	0
$577$	−41.1676	−1.71383	−0.856915	−	0.515457i	$-0.827622\pi$
$577$	−41.1676	−1.71383	−0.856915	+	0.515457i	$0.827622\pi$
$578$	−16.0641	−0.668179
$579$	0	0
$580$	−5.64386	−0.234349
$581$	9.98697	0.414329
$582$	0	0
$583$	8.70489	0.360520
$584$	−0.457360	−0.0189257
$585$	0	0
$586$	22.0158	0.909465
$587$	−46.3411	−1.91270	−0.956352	−	0.292219i	$-0.905607\pi$
$587$	−46.3411	−1.91270	−0.956352	+	0.292219i	$0.905607\pi$
$588$	0	0
$589$	−7.26625	−0.299401
$590$	3.49923	0.144061
$591$	0	0
$592$	−8.27776	−0.340214
$593$	21.7311	0.892389	0.446195	−	0.894936i	$-0.352779\pi$
$593$	21.7311	0.892389	0.446195	+	0.894936i	$0.352779\pi$
$594$	0	0
$595$	−7.88633	−0.323308
$596$	7.10431	0.291004
$597$	0	0
$598$	−29.0751	−1.18897
$599$	28.4744	1.16343	0.581715	−	0.813392i	$-0.302382\pi$
$599$	28.4744	1.16343	0.581715	+	0.813392i	$0.302382\pi$
$600$	0	0
$601$	−19.4791	−0.794571	−0.397285	−	0.917695i	$-0.630048\pi$
$601$	−19.4791	−0.794571	−0.397285	+	0.917695i	$0.630048\pi$
$602$	40.3000	1.64250
$603$	0	0
$604$	1.37907	0.0561134
$605$	7.12612	0.289718
$606$	0	0
$607$	7.81140	0.317055	0.158527	−	0.987355i	$-0.449325\pi$
$607$	7.81140	0.317055	0.158527	+	0.987355i	$0.449325\pi$
$608$	−3.61435	−0.146581
$609$	0	0
$610$	22.7882	0.922666
$611$	−6.00224	−0.242825
$612$	0	0
$613$	−26.5936	−1.07411	−0.537053	−	0.843549i	$-0.680462\pi$
$613$	−26.5936	−1.07411	−0.537053	+	0.843549i	$0.680462\pi$
$614$	6.36070	0.256697
$615$	0	0
$616$	−35.4726	−1.42923
$617$	−16.9063	−0.680624	−0.340312	−	0.940313i	$-0.610533\pi$
$617$	−16.9063	−0.680624	−0.340312	+	0.940313i	$0.610533\pi$
$618$	0	0
$619$	8.05899	0.323918	0.161959	−	0.986797i	$-0.448219\pi$
$619$	8.05899	0.323918	0.161959	+	0.986797i	$0.448219\pi$
$620$	−7.53653	−0.302674
$621$	0	0
$622$	−3.51965	−0.141125
$623$	29.7316	1.19117
$624$	0	0
$625$	−5.11402	−0.204561
$626$	35.9988	1.43880
$627$	0	0
$628$	8.47243	0.338087
$629$	−6.56558	−0.261787
$630$	0	0
$631$	9.58459	0.381557	0.190778	−	0.981633i	$-0.438899\pi$
$631$	9.58459	0.381557	0.190778	+	0.981633i	$0.438899\pi$
$632$	17.7953	0.707860
$633$	0	0
$634$	21.3723	0.848803
$635$	−7.70895	−0.305920
$636$	0	0
$637$	9.00838	0.356925
$638$	24.7837	0.981194
$639$	0	0
$640$	4.11330	0.162592
$641$	−34.7349	−1.37195	−0.685973	−	0.727627i	$-0.740623\pi$
$641$	−34.7349	−1.37195	−0.685973	+	0.727627i	$0.740623\pi$
$642$	0	0
$643$	41.5712	1.63941	0.819705	−	0.572786i	$-0.194138\pi$
$643$	41.5712	1.63941	0.819705	+	0.572786i	$0.194138\pi$
$644$	8.22413	0.324076
$645$	0	0
$646$	2.01782	0.0793900
$647$	32.9689	1.29614	0.648070	−	0.761581i	$-0.275576\pi$
$647$	32.9689	1.29614	0.648070	+	0.761581i	$0.275576\pi$
$648$	0	0
$649$	7.76226	0.304695
$650$	−18.0745	−0.708940
$651$	0	0
$652$	6.88881	0.269787
$653$	−9.99732	−0.391225	−0.195613	−	0.980681i	$-0.562670\pi$
$653$	−9.99732	−0.391225	−0.195613	+	0.980681i	$0.562670\pi$
$654$	0	0
$655$	9.37506	0.366314
$656$	24.2005	0.944872
$657$	0	0
$658$	−3.36090	−0.131021
$659$	−28.8400	−1.12345	−0.561723	−	0.827325i	$-0.689861\pi$
$659$	−28.8400	−1.12345	−0.561723	+	0.827325i	$0.689861\pi$
$660$	0	0
$661$	−9.92805	−0.386156	−0.193078	−	0.981183i	$-0.561847\pi$
$661$	−9.92805	−0.386156	−0.193078	+	0.981183i	$0.561847\pi$
$662$	−10.3116	−0.400774
$663$	0	0
$664$	−10.5472	−0.409312
$665$	−4.50523	−0.174706
$666$	0	0
$667$	−22.8665	−0.885395
$668$	−14.9221	−0.577352
$669$	0	0
$670$	−7.88343	−0.304563
$671$	50.5504	1.95148
$672$	0	0
$673$	21.2951	0.820867	0.410433	−	0.911891i	$-0.365377\pi$
$673$	21.2951	0.820867	0.410433	+	0.911891i	$0.365377\pi$
$674$	−16.1790	−0.623192
$675$	0	0
$676$	−15.4564	−0.594478
$677$	−15.6190	−0.600289	−0.300144	−	0.953894i	$-0.597035\pi$
$677$	−15.6190	−0.600289	−0.300144	+	0.953894i	$0.597035\pi$
$678$	0	0
$679$	−9.47321	−0.363548
$680$	8.32876	0.319393
$681$	0	0
$682$	33.0948	1.26727
$683$	19.4832	0.745502	0.372751	−	0.927931i	$-0.378414\pi$
$683$	19.4832	0.745502	0.372751	+	0.927931i	$0.378414\pi$
$684$	0	0
$685$	25.5545	0.976387
$686$	−18.4821	−0.705650
$687$	0	0
$688$	−26.4635	−1.00891
$689$	13.2236	0.503780
$690$	0	0
$691$	−47.7683	−1.81719	−0.908596	−	0.417677i	$-0.862844\pi$
$691$	−47.7683	−1.81719	−0.908596	+	0.417677i	$0.862844\pi$
$692$	−0.932475	−0.0354474
$693$	0	0
$694$	−29.4907	−1.11945
$695$	17.3092	0.656575
$696$	0	0
$697$	19.1949	0.727057
$698$	26.1560	0.990020
$699$	0	0
$700$	5.11251	0.193235
$701$	40.3514	1.52405	0.762025	−	0.647547i	$-0.224205\pi$
$701$	40.3514	1.52405	0.762025	+	0.647547i	$0.224205\pi$
$702$	0	0
$703$	−3.75073	−0.141461
$704$	33.9022	1.27774
$705$	0	0
$706$	39.4926	1.48632
$707$	−29.5517	−1.11141
$708$	0	0
$709$	−37.9156	−1.42395	−0.711976	−	0.702204i	$-0.752199\pi$
$709$	−37.9156	−1.42395	−0.711976	+	0.702204i	$0.752199\pi$
$710$	−11.4181	−0.428513
$711$	0	0
$712$	−31.3996	−1.17675
$713$	−30.5348	−1.14354
$714$	0	0
$715$	36.6459	1.37048
$716$	−5.89367	−0.220257
$717$	0	0
$718$	36.3091	1.35504
$719$	−32.7005	−1.21952	−0.609762	−	0.792585i	$-0.708735\pi$
$719$	−32.7005	−1.21952	−0.609762	+	0.792585i	$0.708735\pi$
$720$	0	0
$721$	−21.8935	−0.815356
$722$	1.15272	0.0428998
$723$	0	0
$724$	11.9965	0.445845
$725$	−14.2149	−0.527929
$726$	0	0
$727$	7.48787	0.277710	0.138855	−	0.990313i	$-0.455658\pi$
$727$	7.48787	0.277710	0.138855	+	0.990313i	$0.455658\pi$
$728$	−53.8865	−1.99717
$729$	0	0
$730$	0.264565	0.00979201
$731$	−20.9898	−0.776334
$732$	0	0
$733$	−6.17853	−0.228209	−0.114105	−	0.993469i	$-0.536400\pi$
$733$	−6.17853	−0.228209	−0.114105	+	0.993469i	$0.536400\pi$
$734$	22.0325	0.813233
$735$	0	0
$736$	−15.1885	−0.559855
$737$	−17.4876	−0.644164
$738$	0	0
$739$	−6.20120	−0.228115	−0.114057	−	0.993474i	$-0.536385\pi$
$739$	−6.20120	−0.228115	−0.114057	+	0.993474i	$0.536385\pi$
$740$	−3.89024	−0.143008
$741$	0	0
$742$	7.40444	0.271826
$743$	−32.4975	−1.19222	−0.596108	−	0.802904i	$-0.703287\pi$
$743$	−32.4975	−1.19222	−0.596108	+	0.802904i	$0.703287\pi$
$744$	0	0
$745$	−16.3544	−0.599178
$746$	17.7393	0.649481
$747$	0	0
$748$	4.64256	0.169749
$749$	−58.2048	−2.12676
$750$	0	0
$751$	44.0503	1.60742	0.803709	−	0.595022i	$-0.202857\pi$
$751$	44.0503	1.60742	0.803709	+	0.595022i	$0.202857\pi$
$752$	2.20697	0.0804801
$753$	0	0
$754$	37.6489	1.37109
$755$	−3.17466	−0.115538
$756$	0	0
$757$	36.6520	1.33214	0.666070	−	0.745890i	$-0.267975\pi$
$757$	36.6520	1.33214	0.666070	+	0.745890i	$0.267975\pi$
$758$	0.476013	0.0172896
$759$	0	0
$760$	4.75798	0.172590
$761$	5.53854	0.200772	0.100386	−	0.994949i	$-0.467992\pi$
$761$	5.53854	0.200772	0.100386	+	0.994949i	$0.467992\pi$
$762$	0	0
$763$	−12.6853	−0.459239
$764$	−7.18828	−0.260063
$765$	0	0
$766$	2.21182	0.0799165
$767$	11.7917	0.425773
$768$	0	0
$769$	−39.8756	−1.43795	−0.718975	−	0.695036i	$-0.755388\pi$
$769$	−39.8756	−1.43795	−0.718975	+	0.695036i	$0.755388\pi$
$770$	20.5195	0.739473
$771$	0	0
$772$	−12.8550	−0.462662
$773$	7.09596	0.255224	0.127612	−	0.991824i	$-0.459269\pi$
$773$	7.09596	0.255224	0.127612	+	0.991824i	$0.459269\pi$
$774$	0	0
$775$	−18.9819	−0.681849
$776$	10.0047	0.359146
$777$	0	0
$778$	6.45348	0.231368
$779$	10.9655	0.392879
$780$	0	0
$781$	−25.3285	−0.906323
$782$	8.47942	0.303224
$783$	0	0
$784$	−3.31231	−0.118297
$785$	−19.5038	−0.696122
$786$	0	0
$787$	−25.2420	−0.899781	−0.449891	−	0.893084i	$-0.648537\pi$
$787$	−25.2420	−0.899781	−0.449891	+	0.893084i	$0.648537\pi$
$788$	2.97785	0.106082
$789$	0	0
$790$	−10.2939	−0.366241
$791$	20.1497	0.716441
$792$	0	0
$793$	76.7913	2.72694
$794$	−31.4113	−1.11474
$795$	0	0
$796$	17.9025	0.634538
$797$	−36.7639	−1.30224	−0.651121	−	0.758974i	$-0.725701\pi$
$797$	−36.7639	−1.30224	−0.651121	+	0.758974i	$0.725701\pi$
$798$	0	0
$799$	1.75048	0.0619276
$800$	−9.44190	−0.333821
$801$	0	0
$802$	−28.5672	−1.00874
$803$	0.586879	0.0207105
$804$	0	0
$805$	−18.9322	−0.667274
$806$	50.2745	1.77084
$807$	0	0
$808$	31.2096	1.09795
$809$	−37.4359	−1.31618	−0.658089	−	0.752940i	$-0.728635\pi$
$809$	−37.4359	−1.31618	−0.658089	+	0.752940i	$0.728635\pi$
$810$	0	0
$811$	−8.41256	−0.295405	−0.147703	−	0.989032i	$-0.547188\pi$
$811$	−8.41256	−0.295405	−0.147703	+	0.989032i	$0.547188\pi$
$812$	−10.6493	−0.373717
$813$	0	0
$814$	17.0830	0.598760
$815$	−15.8583	−0.555492
$816$	0	0
$817$	−11.9908	−0.419506
$818$	28.0876	0.982060
$819$	0	0
$820$	11.3734	0.397175
$821$	49.7584	1.73658	0.868290	−	0.496057i	$-0.165219\pi$
$821$	49.7584	1.73658	0.868290	+	0.496057i	$0.165219\pi$
$822$	0	0
$823$	−30.8912	−1.07680	−0.538400	−	0.842690i	$-0.680971\pi$
$823$	−30.8912	−1.07680	−0.538400	+	0.842690i	$0.680971\pi$
$824$	23.1217	0.805483
$825$	0	0
$826$	6.60264	0.229735
$827$	−40.7572	−1.41727	−0.708633	−	0.705577i	$-0.750688\pi$
$827$	−40.7572	−1.41727	−0.708633	+	0.705577i	$0.750688\pi$
$828$	0	0
$829$	−25.7975	−0.895986	−0.447993	−	0.894037i	$-0.647861\pi$
$829$	−25.7975	−0.895986	−0.447993	+	0.894037i	$0.647861\pi$
$830$	6.10117	0.211775
$831$	0	0
$832$	51.5009	1.78547
$833$	−2.62719	−0.0910267
$834$	0	0
$835$	34.3511	1.18877
$836$	2.65216	0.0917270
$837$	0	0
$838$	−43.4456	−1.50080
$839$	−8.04905	−0.277884	−0.138942	−	0.990301i	$-0.544370\pi$
$839$	−8.04905	−0.277884	−0.138942	+	0.990301i	$0.544370\pi$
$840$	0	0
$841$	0.609476	0.0210164
$842$	31.4757	1.08472
$843$	0	0
$844$	−14.6950	−0.505824
$845$	35.5813	1.22403
$846$	0	0
$847$	13.4461	0.462014
$848$	−4.86222	−0.166969
$849$	0	0
$850$	5.27122	0.180801
$851$	−15.7616	−0.540300
$852$	0	0
$853$	−4.90565	−0.167966	−0.0839832	−	0.996467i	$-0.526764\pi$
$853$	−4.90565	−0.167966	−0.0839832	+	0.996467i	$0.526764\pi$
$854$	42.9985	1.47138
$855$	0	0
$856$	61.4701	2.10101
$857$	29.1725	0.996512	0.498256	−	0.867030i	$-0.333974\pi$
$857$	29.1725	0.996512	0.498256	+	0.867030i	$0.333974\pi$
$858$	0	0
$859$	−3.44072	−0.117396	−0.0586979	−	0.998276i	$-0.518695\pi$
$859$	−3.44072	−0.117396	−0.0586979	+	0.998276i	$0.518695\pi$
$860$	−12.4369	−0.424093
$861$	0	0
$862$	−27.2055	−0.926625
$863$	1.46252	0.0497848	0.0248924	−	0.999690i	$-0.492076\pi$
$863$	1.46252	0.0497848	0.0248924	+	0.999690i	$0.492076\pi$
$864$	0	0
$865$	2.14659	0.0729863
$866$	−23.8771	−0.811377
$867$	0	0
$868$	−14.2205	−0.482676
$869$	−22.8347	−0.774615
$870$	0	0
$871$	−26.5655	−0.900137
$872$	13.3969	0.453678
$873$	0	0
$874$	4.84405	0.163852
$875$	−34.2954	−1.15939
$876$	0	0
$877$	−15.4807	−0.522745	−0.261372	−	0.965238i	$-0.584175\pi$
$877$	−15.4807	−0.522745	−0.261372	+	0.965238i	$0.584175\pi$
$878$	31.6456	1.06799
$879$	0	0
$880$	−13.4744	−0.454222
$881$	34.5176	1.16293	0.581464	−	0.813572i	$-0.302480\pi$
$881$	34.5176	1.16293	0.581464	+	0.813572i	$0.302480\pi$
$882$	0	0
$883$	38.1114	1.28255	0.641276	−	0.767310i	$-0.278405\pi$
$883$	38.1114	1.28255	0.641276	+	0.767310i	$0.278405\pi$
$884$	7.05254	0.237202
$885$	0	0
$886$	−11.0393	−0.370872
$887$	2.06621	0.0693767	0.0346883	−	0.999398i	$-0.488956\pi$
$887$	2.06621	0.0693767	0.0346883	+	0.999398i	$0.488956\pi$
$888$	0	0
$889$	−14.5459	−0.487853
$890$	18.1635	0.608841
$891$	0	0
$892$	2.59646	0.0869359
$893$	1.00000	0.0334637
$894$	0	0
$895$	13.5674	0.453510
$896$	7.76130	0.259287
$897$	0	0
$898$	5.19644	0.173407
$899$	39.5390	1.31870
$900$	0	0
$901$	−3.85652	−0.128479
$902$	−49.9433	−1.66293
$903$	0	0
$904$	−21.2801	−0.707766
$905$	−27.6163	−0.917997
$906$	0	0
$907$	−42.6653	−1.41668	−0.708338	−	0.705873i	$-0.750555\pi$
$907$	−42.6653	−1.41668	−0.708338	+	0.705873i	$0.750555\pi$
$908$	−4.27171	−0.141762
$909$	0	0
$910$	31.1713	1.03332
$911$	10.1499	0.336282	0.168141	−	0.985763i	$-0.446224\pi$
$911$	10.1499	0.336282	0.168141	+	0.985763i	$0.446224\pi$
$912$	0	0
$913$	13.5341	0.447913
$914$	−43.1508	−1.42730
$915$	0	0
$916$	−8.63213	−0.285214
$917$	17.6896	0.584163
$918$	0	0
$919$	25.5137	0.841621	0.420810	−	0.907149i	$-0.361746\pi$
$919$	25.5137	0.841621	0.420810	+	0.907149i	$0.361746\pi$
$920$	19.9943	0.659194
$921$	0	0
$922$	9.16342	0.301781
$923$	−38.4765	−1.26647
$924$	0	0
$925$	−9.79815	−0.322161
$926$	1.96704	0.0646409
$927$	0	0
$928$	19.6673	0.645612
$929$	−37.7084	−1.23717	−0.618587	−	0.785716i	$-0.712295\pi$
$929$	−37.7084	−1.23717	−0.618587	+	0.785716i	$0.712295\pi$
$930$	0	0
$931$	−1.50084	−0.0491879
$932$	15.0223	0.492072
$933$	0	0
$934$	−28.6476	−0.937379
$935$	−10.6874	−0.349514
$936$	0	0
$937$	−50.3811	−1.64588	−0.822940	−	0.568128i	$-0.807668\pi$
$937$	−50.3811	−1.64588	−0.822940	+	0.568128i	$0.807668\pi$
$938$	−14.8751	−0.485689
$939$	0	0
$940$	1.03720	0.0338296
$941$	−23.9428	−0.780513	−0.390257	−	0.920706i	$-0.627614\pi$
$941$	−23.9428	−0.780513	−0.390257	+	0.920706i	$0.627614\pi$
$942$	0	0
$943$	46.0799	1.50057
$944$	−4.33570	−0.141115
$945$	0	0
$946$	54.6135	1.77564
$947$	−13.2296	−0.429904	−0.214952	−	0.976625i	$-0.568960\pi$
$947$	−13.2296	−0.429904	−0.214952	+	0.976625i	$0.568960\pi$
$948$	0	0
$949$	0.891529	0.0289403
$950$	3.01129	0.0976993
$951$	0	0
$952$	15.7154	0.509338
$953$	34.0710	1.10367	0.551834	−	0.833954i	$-0.313928\pi$
$953$	34.0710	1.10367	0.551834	+	0.833954i	$0.313928\pi$
$954$	0	0
$955$	16.5477	0.535470
$956$	−12.9637	−0.419276
$957$	0	0
$958$	27.0002	0.872337
$959$	48.2183	1.55705
$960$	0	0
$961$	21.7984	0.703174
$962$	25.9509	0.836691
$963$	0	0
$964$	10.1038	0.325421
$965$	29.5927	0.952624
$966$	0	0
$967$	45.8937	1.47584	0.737921	−	0.674887i	$-0.235808\pi$
$967$	45.8937	1.47584	0.737921	+	0.674887i	$0.235808\pi$
$968$	−14.2005	−0.456420
$969$	0	0
$970$	−5.78731	−0.185819
$971$	48.9781	1.57178	0.785890	−	0.618366i	$-0.212205\pi$
$971$	48.9781	1.57178	0.785890	+	0.618366i	$0.212205\pi$
$972$	0	0
$973$	32.6603	1.04704
$974$	12.5067	0.400740
$975$	0	0
$976$	−28.2355	−0.903798
$977$	16.5254	0.528694	0.264347	−	0.964428i	$-0.414844\pi$
$977$	16.5254	0.528694	0.264347	+	0.964428i	$0.414844\pi$
$978$	0	0
$979$	40.2916	1.28772
$980$	−1.55666	−0.0497258
$981$	0	0
$982$	−24.9635	−0.796616
$983$	5.96485	0.190249	0.0951245	−	0.995465i	$-0.469675\pi$
$983$	5.96485	0.190249	0.0951245	+	0.995465i	$0.469675\pi$
$984$	0	0
$985$	−6.85512	−0.218422
$986$	−10.9799	−0.349670
$987$	0	0
$988$	4.02891	0.128177
$989$	−50.3888	−1.60227
$990$	0	0
$991$	−29.5348	−0.938205	−0.469102	−	0.883144i	$-0.655423\pi$
$991$	−29.5348	−0.938205	−0.469102	+	0.883144i	$0.655423\pi$
$992$	26.2628	0.833844
$993$	0	0
$994$	−21.5446	−0.683352
$995$	−41.2122	−1.30652
$996$	0	0
$997$	−0.184853	−0.00585436	−0.00292718	−	0.999996i	$-0.500932\pi$
$997$	−0.184853	−0.00585436	−0.00292718	+	0.999996i	$0.500932\pi$
$998$	33.6388	1.06482
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.o.1.12		18
3.2	odd	2		893.2.a.c.1.7	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.c.1.7	✓	18	3.2	odd	2
8037.2.a.o.1.12		18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+1.15272 q^{2} -0.671235 q^{4} +1.54521 q^{5} +2.91562 q^{7} -3.07919 q^{8} +O(q^{10})\)	\(q+1.15272 q^{2} -0.671235 q^{4} +1.54521 q^{5} +2.91562 q^{7} -3.07919 q^{8} +1.78119 q^{10} +3.95117 q^{11} +6.00224 q^{13} +3.36090 q^{14} -2.20697 q^{16} -1.75048 q^{17} -1.00000 q^{19} -1.03720 q^{20} +4.55460 q^{22} -4.20228 q^{23} -2.61234 q^{25} +6.91890 q^{26} -1.95706 q^{28} +5.44146 q^{29} +7.26625 q^{31} +3.61435 q^{32} -2.01782 q^{34} +4.50523 q^{35} +3.75073 q^{37} -1.15272 q^{38} -4.75798 q^{40} -10.9655 q^{41} +11.9908 q^{43} -2.65216 q^{44} -4.84405 q^{46} -1.00000 q^{47} +1.50084 q^{49} -3.01129 q^{50} -4.02891 q^{52} +2.20312 q^{53} +6.10538 q^{55} -8.97774 q^{56} +6.27248 q^{58} +1.96455 q^{59} +12.7938 q^{61} +8.37596 q^{62} +8.58029 q^{64} +9.27470 q^{65} -4.42593 q^{67} +1.17498 q^{68} +5.19328 q^{70} -6.41036 q^{71} +0.148533 q^{73} +4.32354 q^{74} +0.671235 q^{76} +11.5201 q^{77} -5.77923 q^{79} -3.41023 q^{80} -12.6401 q^{82} +3.42533 q^{83} -2.70486 q^{85} +13.8221 q^{86} -12.1664 q^{88} +10.1974 q^{89} +17.5002 q^{91} +2.82071 q^{92} -1.15272 q^{94} -1.54521 q^{95} -3.24912 q^{97} +1.73005 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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