LMFDB - Embedded newform 8046.2.a.p.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 23 x^{10} + 142 x^{9} + 104 x^{8} - 1302 x^{7} + 607 x^{6} + 4323 x^{5} - 4461 x^{4} + \cdots - 553 $ x^12 - 5x^11 - 23x^10 + 142x^9 + 104x^8 - 1302x^7 + 607x^6 + 4323x^5 - 4461x^4 - 3333x^3 + 4805x^2 - 224*x - 553
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.27289$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.27289 q^{5} +3.85060 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +2.27289 q^{5} +3.85060 q^{7} +1.00000 q^{8} +2.27289 q^{10} -4.73844 q^{11} +3.85466 q^{13} +3.85060 q^{14} +1.00000 q^{16} +3.14894 q^{17} +4.89535 q^{19} +2.27289 q^{20} -4.73844 q^{22} +6.57014 q^{23} +0.166034 q^{25} +3.85466 q^{26} +3.85060 q^{28} +9.39932 q^{29} -0.492008 q^{31} +1.00000 q^{32} +3.14894 q^{34} +8.75199 q^{35} -4.90618 q^{37} +4.89535 q^{38} +2.27289 q^{40} -3.00687 q^{41} -4.16284 q^{43} -4.73844 q^{44} +6.57014 q^{46} +1.86831 q^{47} +7.82712 q^{49} +0.166034 q^{50} +3.85466 q^{52} -1.94505 q^{53} -10.7700 q^{55} +3.85060 q^{56} +9.39932 q^{58} -9.17305 q^{59} -15.1765 q^{61} -0.492008 q^{62} +1.00000 q^{64} +8.76122 q^{65} -14.1127 q^{67} +3.14894 q^{68} +8.75199 q^{70} +11.0457 q^{71} -14.7517 q^{73} -4.90618 q^{74} +4.89535 q^{76} -18.2458 q^{77} -13.8443 q^{79} +2.27289 q^{80} -3.00687 q^{82} +1.47600 q^{83} +7.15721 q^{85} -4.16284 q^{86} -4.73844 q^{88} +14.8032 q^{89} +14.8428 q^{91} +6.57014 q^{92} +1.86831 q^{94} +11.1266 q^{95} -11.4686 q^{97} +7.82712 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.27289	1.01647	0.508234	−	0.861219i	$-0.330299\pi$
$5$	2.27289	1.01647	0.508234	+	0.861219i	$0.330299\pi$
$6$	0	0
$7$	3.85060	1.45539	0.727695	−	0.685901i	$-0.240592\pi$
$7$	3.85060	1.45539	0.727695	+	0.685901i	$0.240592\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.27289	0.718751
$11$	−4.73844	−1.42869	−0.714346	−	0.699792i	$-0.753276\pi$
$11$	−4.73844	−1.42869	−0.714346	+	0.699792i	$0.753276\pi$
$12$	0	0
$13$	3.85466	1.06909	0.534545	−	0.845140i	$-0.320483\pi$
$13$	3.85466	1.06909	0.534545	+	0.845140i	$0.320483\pi$
$14$	3.85060	1.02912
$15$	0	0
$16$	1.00000	0.250000
$17$	3.14894	0.763731	0.381865	−	0.924218i	$-0.375282\pi$
$17$	3.14894	0.763731	0.381865	+	0.924218i	$0.375282\pi$
$18$	0	0
$19$	4.89535	1.12307	0.561535	−	0.827453i	$-0.310211\pi$
$19$	4.89535	1.12307	0.561535	+	0.827453i	$0.310211\pi$
$20$	2.27289	0.508234
$21$	0	0
$22$	−4.73844	−1.01024
$23$	6.57014	1.36997	0.684985	−	0.728558i	$-0.259809\pi$
$23$	6.57014	1.36997	0.684985	+	0.728558i	$0.259809\pi$
$24$	0	0
$25$	0.166034	0.0332069
$26$	3.85466	0.755961
$27$	0	0
$28$	3.85060	0.727695
$29$	9.39932	1.74541	0.872705	−	0.488248i	$-0.162364\pi$
$29$	9.39932	1.74541	0.872705	+	0.488248i	$0.162364\pi$
$30$	0	0
$31$	−0.492008	−0.0883672	−0.0441836	−	0.999023i	$-0.514069\pi$
$31$	−0.492008	−0.0883672	−0.0441836	+	0.999023i	$0.514069\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.14894	0.540039
$35$	8.75199	1.47936
$36$	0	0
$37$	−4.90618	−0.806571	−0.403285	−	0.915074i	$-0.632132\pi$
$37$	−4.90618	−0.806571	−0.403285	+	0.915074i	$0.632132\pi$
$38$	4.89535	0.794130
$39$	0	0
$40$	2.27289	0.359376
$41$	−3.00687	−0.469594	−0.234797	−	0.972044i	$-0.575442\pi$
$41$	−3.00687	−0.469594	−0.234797	+	0.972044i	$0.575442\pi$
$42$	0	0
$43$	−4.16284	−0.634828	−0.317414	−	0.948287i	$-0.602814\pi$
$43$	−4.16284	−0.634828	−0.317414	+	0.948287i	$0.602814\pi$
$44$	−4.73844	−0.714346
$45$	0	0
$46$	6.57014	0.968714
$47$	1.86831	0.272521	0.136260	−	0.990673i	$-0.456492\pi$
$47$	1.86831	0.272521	0.136260	+	0.990673i	$0.456492\pi$
$48$	0	0
$49$	7.82712	1.11816
$50$	0.166034	0.0234808
$51$	0	0
$52$	3.85466	0.534545
$53$	−1.94505	−0.267173	−0.133586	−	0.991037i	$-0.542649\pi$
$53$	−1.94505	−0.267173	−0.133586	+	0.991037i	$0.542649\pi$
$54$	0	0
$55$	−10.7700	−1.45222
$56$	3.85060	0.514558
$57$	0	0
$58$	9.39932	1.23419
$59$	−9.17305	−1.19423	−0.597114	−	0.802156i	$-0.703686\pi$
$59$	−9.17305	−1.19423	−0.597114	+	0.802156i	$0.703686\pi$
$60$	0	0
$61$	−15.1765	−1.94316	−0.971578	−	0.236718i	$-0.923928\pi$
$61$	−15.1765	−1.94316	−0.971578	+	0.236718i	$0.923928\pi$
$62$	−0.492008	−0.0624851
$63$	0	0
$64$	1.00000	0.125000
$65$	8.76122	1.08670
$66$	0	0
$67$	−14.1127	−1.72414	−0.862069	−	0.506790i	$-0.830832\pi$
$67$	−14.1127	−1.72414	−0.862069	+	0.506790i	$0.830832\pi$
$68$	3.14894	0.381865
$69$	0	0
$70$	8.75199	1.04606
$71$	11.0457	1.31088	0.655442	−	0.755246i	$-0.272483\pi$
$71$	11.0457	1.31088	0.655442	+	0.755246i	$0.272483\pi$
$72$	0	0
$73$	−14.7517	−1.72656	−0.863278	−	0.504728i	$-0.831593\pi$
$73$	−14.7517	−1.72656	−0.863278	+	0.504728i	$0.831593\pi$
$74$	−4.90618	−0.570331
$75$	0	0
$76$	4.89535	0.561535
$77$	−18.2458	−2.07931
$78$	0	0
$79$	−13.8443	−1.55760	−0.778800	−	0.627272i	$-0.784171\pi$
$79$	−13.8443	−1.55760	−0.778800	+	0.627272i	$0.784171\pi$
$80$	2.27289	0.254117
$81$	0	0
$82$	−3.00687	−0.332053
$83$	1.47600	0.162012	0.0810062	−	0.996714i	$-0.474187\pi$
$83$	1.47600	0.162012	0.0810062	+	0.996714i	$0.474187\pi$
$84$	0	0
$85$	7.15721	0.776308
$86$	−4.16284	−0.448891
$87$	0	0
$88$	−4.73844	−0.505119
$89$	14.8032	1.56913	0.784566	−	0.620045i	$-0.212886\pi$
$89$	14.8032	1.56913	0.784566	+	0.620045i	$0.212886\pi$
$90$	0	0
$91$	14.8428	1.55594
$92$	6.57014	0.684985
$93$	0	0
$94$	1.86831	0.192701
$95$	11.1266	1.14156
$96$	0	0
$97$	−11.4686	−1.16446	−0.582228	−	0.813025i	$-0.697819\pi$
$97$	−11.4686	−1.16446	−0.582228	+	0.813025i	$0.697819\pi$
$98$	7.82712	0.790658
$99$	0	0
$100$	0.166034	0.0166034
$101$	13.2799	1.32140	0.660701	−	0.750649i	$-0.270259\pi$
$101$	13.2799	1.32140	0.660701	+	0.750649i	$0.270259\pi$
$102$	0	0
$103$	9.35943	0.922212	0.461106	−	0.887345i	$-0.347453\pi$
$103$	9.35943	0.922212	0.461106	+	0.887345i	$0.347453\pi$
$104$	3.85466	0.377980
$105$	0	0
$106$	−1.94505	−0.188920
$107$	11.3743	1.09960	0.549799	−	0.835297i	$-0.314704\pi$
$107$	11.3743	1.09960	0.549799	+	0.835297i	$0.314704\pi$
$108$	0	0
$109$	11.6123	1.11225	0.556127	−	0.831098i	$-0.312287\pi$
$109$	11.6123	1.11225	0.556127	+	0.831098i	$0.312287\pi$
$110$	−10.7700	−1.02687
$111$	0	0
$112$	3.85060	0.363847
$113$	−16.2630	−1.52989	−0.764947	−	0.644093i	$-0.777235\pi$
$113$	−16.2630	−1.52989	−0.764947	+	0.644093i	$0.777235\pi$
$114$	0	0
$115$	14.9332	1.39253
$116$	9.39932	0.872705
$117$	0	0
$118$	−9.17305	−0.844447
$119$	12.1253	1.11153
$120$	0	0
$121$	11.4528	1.04116
$122$	−15.1765	−1.37402
$123$	0	0
$124$	−0.492008	−0.0441836
$125$	−10.9871	−0.982714
$126$	0	0
$127$	16.6363	1.47623	0.738116	−	0.674673i	$-0.235716\pi$
$127$	16.6363	1.47623	0.738116	+	0.674673i	$0.235716\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	8.76122	0.768410
$131$	20.2513	1.76936	0.884680	−	0.466198i	$-0.154377\pi$
$131$	20.2513	1.76936	0.884680	+	0.466198i	$0.154377\pi$
$132$	0	0
$133$	18.8500	1.63450
$134$	−14.1127	−1.21915
$135$	0	0
$136$	3.14894	0.270020
$137$	−10.6034	−0.905911	−0.452956	−	0.891533i	$-0.649630\pi$
$137$	−10.6034	−0.905911	−0.452956	+	0.891533i	$0.649630\pi$
$138$	0	0
$139$	−20.2526	−1.71780	−0.858900	−	0.512144i	$-0.828852\pi$
$139$	−20.2526	−1.71780	−0.858900	+	0.512144i	$0.828852\pi$
$140$	8.75199	0.739679
$141$	0	0
$142$	11.0457	0.926934
$143$	−18.2651	−1.52740
$144$	0	0
$145$	21.3636	1.77415
$146$	−14.7517	−1.22086
$147$	0	0
$148$	−4.90618	−0.403285
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	9.67168	0.787070	0.393535	−	0.919310i	$-0.371252\pi$
$151$	9.67168	0.787070	0.393535	+	0.919310i	$0.371252\pi$
$152$	4.89535	0.397065
$153$	0	0
$154$	−18.2458	−1.47029
$155$	−1.11828	−0.0898225
$156$	0	0
$157$	−0.782637	−0.0624612	−0.0312306	−	0.999512i	$-0.509943\pi$
$157$	−0.782637	−0.0624612	−0.0312306	+	0.999512i	$0.509943\pi$
$158$	−13.8443	−1.10139
$159$	0	0
$160$	2.27289	0.179688
$161$	25.2990	1.99384
$162$	0	0
$163$	17.1474	1.34309	0.671545	−	0.740964i	$-0.265631\pi$
$163$	17.1474	1.34309	0.671545	+	0.740964i	$0.265631\pi$
$164$	−3.00687	−0.234797
$165$	0	0
$166$	1.47600	0.114560
$167$	−8.70660	−0.673737	−0.336868	−	0.941552i	$-0.609368\pi$
$167$	−8.70660	−0.673737	−0.336868	+	0.941552i	$0.609368\pi$
$168$	0	0
$169$	1.85840	0.142954
$170$	7.15721	0.548933
$171$	0	0
$172$	−4.16284	−0.317414
$173$	2.86163	0.217566	0.108783	−	0.994066i	$-0.465305\pi$
$173$	2.86163	0.217566	0.108783	+	0.994066i	$0.465305\pi$
$174$	0	0
$175$	0.639332	0.0483289
$176$	−4.73844	−0.357173
$177$	0	0
$178$	14.8032	1.10954
$179$	−5.65183	−0.422437	−0.211219	−	0.977439i	$-0.567743\pi$
$179$	−5.65183	−0.422437	−0.211219	+	0.977439i	$0.567743\pi$
$180$	0	0
$181$	−7.26999	−0.540374	−0.270187	−	0.962808i	$-0.587086\pi$
$181$	−7.26999	−0.540374	−0.270187	+	0.962808i	$0.587086\pi$
$182$	14.8428	1.10022
$183$	0	0
$184$	6.57014	0.484357
$185$	−11.1512	−0.819853
$186$	0	0
$187$	−14.9211	−1.09114
$188$	1.86831	0.136260
$189$	0	0
$190$	11.1266	0.807208
$191$	−10.1345	−0.733308	−0.366654	−	0.930357i	$-0.619497\pi$
$191$	−10.1345	−0.733308	−0.366654	+	0.930357i	$0.619497\pi$
$192$	0	0
$193$	−18.4300	−1.32662	−0.663310	−	0.748345i	$-0.730849\pi$
$193$	−18.4300	−1.32662	−0.663310	+	0.748345i	$0.730849\pi$
$194$	−11.4686	−0.823395
$195$	0	0
$196$	7.82712	0.559080
$197$	−16.4840	−1.17444	−0.587218	−	0.809429i	$-0.699777\pi$
$197$	−16.4840	−1.17444	−0.587218	+	0.809429i	$0.699777\pi$
$198$	0	0
$199$	−4.38822	−0.311073	−0.155536	−	0.987830i	$-0.549711\pi$
$199$	−4.38822	−0.311073	−0.155536	+	0.987830i	$0.549711\pi$
$200$	0.166034	0.0117404
$201$	0	0
$202$	13.2799	0.934372
$203$	36.1930	2.54025
$204$	0	0
$205$	−6.83428	−0.477327
$206$	9.35943	0.652102
$207$	0	0
$208$	3.85466	0.267273
$209$	−23.1963	−1.60452
$210$	0	0
$211$	−9.44934	−0.650520	−0.325260	−	0.945625i	$-0.605452\pi$
$211$	−9.44934	−0.650520	−0.325260	+	0.945625i	$0.605452\pi$
$212$	−1.94505	−0.133586
$213$	0	0
$214$	11.3743	0.777533
$215$	−9.46169	−0.645282
$216$	0	0
$217$	−1.89453	−0.128609
$218$	11.6123	0.786482
$219$	0	0
$220$	−10.7700	−0.726110
$221$	12.1381	0.816497
$222$	0	0
$223$	−5.39202	−0.361076	−0.180538	−	0.983568i	$-0.557784\pi$
$223$	−5.39202	−0.361076	−0.180538	+	0.983568i	$0.557784\pi$
$224$	3.85060	0.257279
$225$	0	0
$226$	−16.2630	−1.08180
$227$	25.4926	1.69201	0.846003	−	0.533178i	$-0.179002\pi$
$227$	25.4926	1.69201	0.846003	+	0.533178i	$0.179002\pi$
$228$	0	0
$229$	23.7053	1.56649	0.783246	−	0.621712i	$-0.213563\pi$
$229$	23.7053	1.56649	0.783246	+	0.621712i	$0.213563\pi$
$230$	14.9332	0.984667
$231$	0	0
$232$	9.39932	0.617096
$233$	−22.3704	−1.46553	−0.732767	−	0.680480i	$-0.761771\pi$
$233$	−22.3704	−1.46553	−0.732767	+	0.680480i	$0.761771\pi$
$234$	0	0
$235$	4.24646	0.277009
$236$	−9.17305	−0.597114
$237$	0	0
$238$	12.1253	0.785968
$239$	0.194267	0.0125661	0.00628304	−	0.999980i	$-0.498000\pi$
$239$	0.194267	0.0125661	0.00628304	+	0.999980i	$0.498000\pi$
$240$	0	0
$241$	24.1769	1.55737	0.778685	−	0.627415i	$-0.215887\pi$
$241$	24.1769	1.55737	0.778685	+	0.627415i	$0.215887\pi$
$242$	11.4528	0.736214
$243$	0	0
$244$	−15.1765	−0.971578
$245$	17.7902	1.13657
$246$	0	0
$247$	18.8699	1.20066
$248$	−0.492008	−0.0312425
$249$	0	0
$250$	−10.9871	−0.694884
$251$	9.20467	0.580994	0.290497	−	0.956876i	$-0.406179\pi$
$251$	9.20467	0.580994	0.290497	+	0.956876i	$0.406179\pi$
$252$	0	0
$253$	−31.1322	−1.95727
$254$	16.6363	1.04385
$255$	0	0
$256$	1.00000	0.0625000
$257$	−30.5416	−1.90514	−0.952568	−	0.304327i	$-0.901569\pi$
$257$	−30.5416	−1.90514	−0.952568	+	0.304327i	$0.901569\pi$
$258$	0	0
$259$	−18.8917	−1.17387
$260$	8.76122	0.543348
$261$	0	0
$262$	20.2513	1.25113
$263$	−14.1009	−0.869501	−0.434750	−	0.900551i	$-0.643163\pi$
$263$	−14.1009	−0.869501	−0.434750	+	0.900551i	$0.643163\pi$
$264$	0	0
$265$	−4.42088	−0.271573
$266$	18.8500	1.15577
$267$	0	0
$268$	−14.1127	−0.862069
$269$	−3.18315	−0.194080	−0.0970402	−	0.995280i	$-0.530938\pi$
$269$	−3.18315	−0.194080	−0.0970402	+	0.995280i	$0.530938\pi$
$270$	0	0
$271$	2.86180	0.173842	0.0869210	−	0.996215i	$-0.472297\pi$
$271$	2.86180	0.173842	0.0869210	+	0.996215i	$0.472297\pi$
$272$	3.14894	0.190933
$273$	0	0
$274$	−10.6034	−0.640576
$275$	−0.786743	−0.0474424
$276$	0	0
$277$	20.2973	1.21955	0.609774	−	0.792576i	$-0.291260\pi$
$277$	20.2973	1.21955	0.609774	+	0.792576i	$0.291260\pi$
$278$	−20.2526	−1.21467
$279$	0	0
$280$	8.75199	0.523032
$281$	18.8798	1.12628	0.563138	−	0.826363i	$-0.309594\pi$
$281$	18.8798	1.12628	0.563138	+	0.826363i	$0.309594\pi$
$282$	0	0
$283$	−8.95544	−0.532345	−0.266173	−	0.963925i	$-0.585759\pi$
$283$	−8.95544	−0.532345	−0.266173	+	0.963925i	$0.585759\pi$
$284$	11.0457	0.655442
$285$	0	0
$286$	−18.2651	−1.08004
$287$	−11.5782	−0.683442
$288$	0	0
$289$	−7.08416	−0.416715
$290$	21.3636	1.25452
$291$	0	0
$292$	−14.7517	−0.863278
$293$	15.5104	0.906128	0.453064	−	0.891478i	$-0.350331\pi$
$293$	15.5104	0.906128	0.453064	+	0.891478i	$0.350331\pi$
$294$	0	0
$295$	−20.8493	−1.21390
$296$	−4.90618	−0.285166
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	25.3257	1.46462
$300$	0	0
$301$	−16.0294	−0.923922
$302$	9.67168	0.556543
$303$	0	0
$304$	4.89535	0.280767
$305$	−34.4946	−1.97516
$306$	0	0
$307$	5.36823	0.306381	0.153191	−	0.988197i	$-0.451045\pi$
$307$	5.36823	0.306381	0.153191	+	0.988197i	$0.451045\pi$
$308$	−18.2458	−1.03965
$309$	0	0
$310$	−1.11828	−0.0635141
$311$	−14.8173	−0.840209	−0.420105	−	0.907476i	$-0.638007\pi$
$311$	−14.8173	−0.840209	−0.420105	+	0.907476i	$0.638007\pi$
$312$	0	0
$313$	10.0739	0.569411	0.284706	−	0.958615i	$-0.408104\pi$
$313$	10.0739	0.569411	0.284706	+	0.958615i	$0.408104\pi$
$314$	−0.782637	−0.0441668
$315$	0	0
$316$	−13.8443	−0.778800
$317$	6.31050	0.354433	0.177217	−	0.984172i	$-0.443291\pi$
$317$	6.31050	0.354433	0.177217	+	0.984172i	$0.443291\pi$
$318$	0	0
$319$	−44.5381	−2.49365
$320$	2.27289	0.127058
$321$	0	0
$322$	25.2990	1.40986
$323$	15.4152	0.857723
$324$	0	0
$325$	0.640006	0.0355011
$326$	17.1474	0.949708
$327$	0	0
$328$	−3.00687	−0.166026
$329$	7.19411	0.396624
$330$	0	0
$331$	−22.6231	−1.24348	−0.621740	−	0.783224i	$-0.713574\pi$
$331$	−22.6231	−1.24348	−0.621740	+	0.783224i	$0.713574\pi$
$332$	1.47600	0.0810062
$333$	0	0
$334$	−8.70660	−0.476404
$335$	−32.0766	−1.75253
$336$	0	0
$337$	5.87625	0.320100	0.160050	−	0.987109i	$-0.448834\pi$
$337$	5.87625	0.320100	0.160050	+	0.987109i	$0.448834\pi$
$338$	1.85840	0.101084
$339$	0	0
$340$	7.15721	0.388154
$341$	2.33135	0.126250
$342$	0	0
$343$	3.18490	0.171969
$344$	−4.16284	−0.224446
$345$	0	0
$346$	2.86163	0.153842
$347$	35.2993	1.89497	0.947484	−	0.319803i	$-0.103617\pi$
$347$	35.2993	1.89497	0.947484	+	0.319803i	$0.103617\pi$
$348$	0	0
$349$	−19.4442	−1.04083	−0.520413	−	0.853915i	$-0.674222\pi$
$349$	−19.4442	−1.04083	−0.520413	+	0.853915i	$0.674222\pi$
$350$	0.639332	0.0341737
$351$	0	0
$352$	−4.73844	−0.252560
$353$	−9.38324	−0.499420	−0.249710	−	0.968321i	$-0.580335\pi$
$353$	−9.38324	−0.499420	−0.249710	+	0.968321i	$0.580335\pi$
$354$	0	0
$355$	25.1057	1.33247
$356$	14.8032	0.784566
$357$	0	0
$358$	−5.65183	−0.298708
$359$	1.80878	0.0954638	0.0477319	−	0.998860i	$-0.484801\pi$
$359$	1.80878	0.0954638	0.0477319	+	0.998860i	$0.484801\pi$
$360$	0	0
$361$	4.96444	0.261286
$362$	−7.26999	−0.382102
$363$	0	0
$364$	14.8428	0.777972
$365$	−33.5290	−1.75499
$366$	0	0
$367$	25.0204	1.30605	0.653027	−	0.757335i	$-0.273499\pi$
$367$	25.0204	1.30605	0.653027	+	0.757335i	$0.273499\pi$
$368$	6.57014	0.342492
$369$	0	0
$370$	−11.1512	−0.579724
$371$	−7.48960	−0.388841
$372$	0	0
$373$	6.60049	0.341760	0.170880	−	0.985292i	$-0.445339\pi$
$373$	6.60049	0.341760	0.170880	+	0.985292i	$0.445339\pi$
$374$	−14.9211	−0.771550
$375$	0	0
$376$	1.86831	0.0963507
$377$	36.2312	1.86600
$378$	0	0
$379$	20.8003	1.06844	0.534221	−	0.845345i	$-0.320605\pi$
$379$	20.8003	1.06844	0.534221	+	0.845345i	$0.320605\pi$
$380$	11.1266	0.570782
$381$	0	0
$382$	−10.1345	−0.518527
$383$	14.7037	0.751323	0.375661	−	0.926757i	$-0.377416\pi$
$383$	14.7037	0.751323	0.375661	+	0.926757i	$0.377416\pi$
$384$	0	0
$385$	−41.4708	−2.11355
$386$	−18.4300	−0.938061
$387$	0	0
$388$	−11.4686	−0.582228
$389$	−7.34324	−0.372317	−0.186159	−	0.982520i	$-0.559604\pi$
$389$	−7.34324	−0.372317	−0.186159	+	0.982520i	$0.559604\pi$
$390$	0	0
$391$	20.6890	1.04629
$392$	7.82712	0.395329
$393$	0	0
$394$	−16.4840	−0.830452
$395$	−31.4665	−1.58325
$396$	0	0
$397$	37.4834	1.88124	0.940619	−	0.339463i	$-0.110245\pi$
$397$	37.4834	1.88124	0.940619	+	0.339463i	$0.110245\pi$
$398$	−4.38822	−0.219962
$399$	0	0
$400$	0.166034	0.00830172
$401$	37.9835	1.89681	0.948404	−	0.317065i	$-0.102697\pi$
$401$	37.9835	1.89681	0.948404	+	0.317065i	$0.102697\pi$
$402$	0	0
$403$	−1.89652	−0.0944726
$404$	13.2799	0.660701
$405$	0	0
$406$	36.1930	1.79623
$407$	23.2476	1.15234
$408$	0	0
$409$	8.15511	0.403244	0.201622	−	0.979463i	$-0.435379\pi$
$409$	8.15511	0.403244	0.201622	+	0.979463i	$0.435379\pi$
$410$	−6.83428	−0.337521
$411$	0	0
$412$	9.35943	0.461106
$413$	−35.3217	−1.73807
$414$	0	0
$415$	3.35480	0.164680
$416$	3.85466	0.188990
$417$	0	0
$418$	−23.1963	−1.13457
$419$	6.56704	0.320821	0.160411	−	0.987050i	$-0.448718\pi$
$419$	6.56704	0.320821	0.160411	+	0.987050i	$0.448718\pi$
$420$	0	0
$421$	−2.38560	−0.116267	−0.0581336	−	0.998309i	$-0.518515\pi$
$421$	−2.38560	−0.116267	−0.0581336	+	0.998309i	$0.518515\pi$
$422$	−9.44934	−0.459987
$423$	0	0
$424$	−1.94505	−0.0944599
$425$	0.522833	0.0253611
$426$	0	0
$427$	−58.4388	−2.82805
$428$	11.3743	0.549799
$429$	0	0
$430$	−9.46169	−0.456283
$431$	5.91578	0.284953	0.142477	−	0.989798i	$-0.454493\pi$
$431$	5.91578	0.284953	0.142477	+	0.989798i	$0.454493\pi$
$432$	0	0
$433$	−18.6739	−0.897412	−0.448706	−	0.893680i	$-0.648115\pi$
$433$	−18.6739	−0.897412	−0.448706	+	0.893680i	$0.648115\pi$
$434$	−1.89453	−0.0909401
$435$	0	0
$436$	11.6123	0.556127
$437$	32.1631	1.53857
$438$	0	0
$439$	−36.4850	−1.74133	−0.870667	−	0.491873i	$-0.836312\pi$
$439$	−36.4850	−1.74133	−0.870667	+	0.491873i	$0.836312\pi$
$440$	−10.7700	−0.513437
$441$	0	0
$442$	12.1381	0.577351
$443$	0.771418	0.0366512	0.0183256	−	0.999832i	$-0.494166\pi$
$443$	0.771418	0.0366512	0.0183256	+	0.999832i	$0.494166\pi$
$444$	0	0
$445$	33.6460	1.59497
$446$	−5.39202	−0.255320
$447$	0	0
$448$	3.85060	0.181924
$449$	4.71625	0.222574	0.111287	−	0.993788i	$-0.464503\pi$
$449$	4.71625	0.222574	0.111287	+	0.993788i	$0.464503\pi$
$450$	0	0
$451$	14.2479	0.670905
$452$	−16.2630	−0.764947
$453$	0	0
$454$	25.4926	1.19643
$455$	33.7360	1.58157
$456$	0	0
$457$	−39.2811	−1.83749	−0.918747	−	0.394847i	$-0.870797\pi$
$457$	−39.2811	−1.83749	−0.918747	+	0.394847i	$0.870797\pi$
$458$	23.7053	1.10768
$459$	0	0
$460$	14.9332	0.696265
$461$	−16.5258	−0.769684	−0.384842	−	0.922983i	$-0.625744\pi$
$461$	−16.5258	−0.769684	−0.384842	+	0.922983i	$0.625744\pi$
$462$	0	0
$463$	17.3522	0.806425	0.403212	−	0.915106i	$-0.367894\pi$
$463$	17.3522	0.806425	0.403212	+	0.915106i	$0.367894\pi$
$464$	9.39932	0.436353
$465$	0	0
$466$	−22.3704	−1.03629
$467$	12.8545	0.594834	0.297417	−	0.954748i	$-0.403875\pi$
$467$	12.8545	0.594834	0.297417	+	0.954748i	$0.403875\pi$
$468$	0	0
$469$	−54.3423	−2.50929
$470$	4.24646	0.195875
$471$	0	0
$472$	−9.17305	−0.422224
$473$	19.7254	0.906974
$474$	0	0
$475$	0.812796	0.0372936
$476$	12.1253	0.555763
$477$	0	0
$478$	0.194267	0.00888556
$479$	20.4482	0.934302	0.467151	−	0.884177i	$-0.345280\pi$
$479$	20.4482	0.934302	0.467151	+	0.884177i	$0.345280\pi$
$480$	0	0
$481$	−18.9116	−0.862297
$482$	24.1769	1.10123
$483$	0	0
$484$	11.4528	0.520582
$485$	−26.0668	−1.18363
$486$	0	0
$487$	−20.3997	−0.924400	−0.462200	−	0.886776i	$-0.652940\pi$
$487$	−20.3997	−0.924400	−0.462200	+	0.886776i	$0.652940\pi$
$488$	−15.1765	−0.687010
$489$	0	0
$490$	17.7902	0.803679
$491$	−17.5220	−0.790756	−0.395378	−	0.918518i	$-0.629386\pi$
$491$	−17.5220	−0.790756	−0.395378	+	0.918518i	$0.629386\pi$
$492$	0	0
$493$	29.5979	1.33302
$494$	18.8699	0.848997
$495$	0	0
$496$	−0.492008	−0.0220918
$497$	42.5326	1.90785
$498$	0	0
$499$	−23.2421	−1.04046	−0.520231	−	0.854026i	$-0.674154\pi$
$499$	−23.2421	−1.04046	−0.520231	+	0.854026i	$0.674154\pi$
$500$	−10.9871	−0.491357
$501$	0	0
$502$	9.20467	0.410825
$503$	34.1711	1.52361	0.761807	−	0.647804i	$-0.224313\pi$
$503$	34.1711	1.52361	0.761807	+	0.647804i	$0.224313\pi$
$504$	0	0
$505$	30.1838	1.34316
$506$	−31.1322	−1.38400
$507$	0	0
$508$	16.6363	0.738116
$509$	−26.6276	−1.18025	−0.590123	−	0.807314i	$-0.700921\pi$
$509$	−26.6276	−1.18025	−0.590123	+	0.807314i	$0.700921\pi$
$510$	0	0
$511$	−56.8029	−2.51281
$512$	1.00000	0.0441942
$513$	0	0
$514$	−30.5416	−1.34713
$515$	21.2730	0.937399
$516$	0	0
$517$	−8.85286	−0.389349
$518$	−18.8917	−0.830055
$519$	0	0
$520$	8.76122	0.384205
$521$	−4.73769	−0.207562	−0.103781	−	0.994600i	$-0.533094\pi$
$521$	−4.73769	−0.207562	−0.103781	+	0.994600i	$0.533094\pi$
$522$	0	0
$523$	−32.3949	−1.41653	−0.708266	−	0.705946i	$-0.750522\pi$
$523$	−32.3949	−1.41653	−0.708266	+	0.705946i	$0.750522\pi$
$524$	20.2513	0.884680
$525$	0	0
$526$	−14.1009	−0.614830
$527$	−1.54931	−0.0674888
$528$	0	0
$529$	20.1668	0.876816
$530$	−4.42088	−0.192031
$531$	0	0
$532$	18.8500	0.817252
$533$	−11.5904	−0.502038
$534$	0	0
$535$	25.8526	1.11771
$536$	−14.1127	−0.609575
$537$	0	0
$538$	−3.18315	−0.137236
$539$	−37.0883	−1.59751
$540$	0	0
$541$	−9.54877	−0.410534	−0.205267	−	0.978706i	$-0.565806\pi$
$541$	−9.54877	−0.410534	−0.205267	+	0.978706i	$0.565806\pi$
$542$	2.86180	0.122925
$543$	0	0
$544$	3.14894	0.135010
$545$	26.3934	1.13057
$546$	0	0
$547$	15.0536	0.643645	0.321822	−	0.946800i	$-0.395705\pi$
$547$	15.0536	0.643645	0.321822	+	0.946800i	$0.395705\pi$
$548$	−10.6034	−0.452956
$549$	0	0
$550$	−0.786743	−0.0335469
$551$	46.0129	1.96022
$552$	0	0
$553$	−53.3087	−2.26692
$554$	20.2973	0.862350
$555$	0	0
$556$	−20.2526	−0.858900
$557$	−31.6503	−1.34106	−0.670532	−	0.741880i	$-0.733934\pi$
$557$	−31.6503	−1.34106	−0.670532	+	0.741880i	$0.733934\pi$
$558$	0	0
$559$	−16.0463	−0.678688
$560$	8.75199	0.369839
$561$	0	0
$562$	18.8798	0.796397
$563$	15.8801	0.669267	0.334633	−	0.942348i	$-0.391387\pi$
$563$	15.8801	0.669267	0.334633	+	0.942348i	$0.391387\pi$
$564$	0	0
$565$	−36.9640	−1.55509
$566$	−8.95544	−0.376425
$567$	0	0
$568$	11.0457	0.463467
$569$	−28.1005	−1.17803	−0.589016	−	0.808121i	$-0.700485\pi$
$569$	−28.1005	−1.17803	−0.589016	+	0.808121i	$0.700485\pi$
$570$	0	0
$571$	26.0755	1.09123	0.545613	−	0.838038i	$-0.316297\pi$
$571$	26.0755	1.09123	0.545613	+	0.838038i	$0.316297\pi$
$572$	−18.2651	−0.763701
$573$	0	0
$574$	−11.5782	−0.483266
$575$	1.09087	0.0454924
$576$	0	0
$577$	24.7241	1.02928	0.514639	−	0.857407i	$-0.327926\pi$
$577$	24.7241	1.02928	0.514639	+	0.857407i	$0.327926\pi$
$578$	−7.08416	−0.294662
$579$	0	0
$580$	21.3636	0.887077
$581$	5.68350	0.235791
$582$	0	0
$583$	9.21649	0.381708
$584$	−14.7517	−0.610430
$585$	0	0
$586$	15.5104	0.640729
$587$	20.2420	0.835475	0.417738	−	0.908568i	$-0.362823\pi$
$587$	20.2420	0.835475	0.417738	+	0.908568i	$0.362823\pi$
$588$	0	0
$589$	−2.40855	−0.0992426
$590$	−20.8493	−0.858354
$591$	0	0
$592$	−4.90618	−0.201643
$593$	11.6428	0.478114	0.239057	−	0.971006i	$-0.423162\pi$
$593$	11.6428	0.478114	0.239057	+	0.971006i	$0.423162\pi$
$594$	0	0
$595$	27.5595	1.12983
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	25.3257	1.03564
$599$	−33.9407	−1.38678	−0.693389	−	0.720564i	$-0.743883\pi$
$599$	−33.9407	−1.38678	−0.693389	+	0.720564i	$0.743883\pi$
$600$	0	0
$601$	−33.7093	−1.37503	−0.687515	−	0.726170i	$-0.741299\pi$
$601$	−33.7093	−1.37503	−0.687515	+	0.726170i	$0.741299\pi$
$602$	−16.0294	−0.653312
$603$	0	0
$604$	9.67168	0.393535
$605$	26.0310	1.05831
$606$	0	0
$607$	35.3991	1.43681	0.718403	−	0.695627i	$-0.244873\pi$
$607$	35.3991	1.43681	0.718403	+	0.695627i	$0.244873\pi$
$608$	4.89535	0.198533
$609$	0	0
$610$	−34.4946	−1.39665
$611$	7.20169	0.291349
$612$	0	0
$613$	16.3409	0.660003	0.330002	−	0.943980i	$-0.392951\pi$
$613$	16.3409	0.660003	0.330002	+	0.943980i	$0.392951\pi$
$614$	5.36823	0.216644
$615$	0	0
$616$	−18.2458	−0.735145
$617$	−22.5999	−0.909837	−0.454918	−	0.890533i	$-0.650332\pi$
$617$	−22.5999	−0.909837	−0.454918	+	0.890533i	$0.650332\pi$
$618$	0	0
$619$	−1.83406	−0.0737170	−0.0368585	−	0.999320i	$-0.511735\pi$
$619$	−1.83406	−0.0737170	−0.0368585	+	0.999320i	$0.511735\pi$
$620$	−1.11828	−0.0449112
$621$	0	0
$622$	−14.8173	−0.594118
$623$	57.0010	2.28370
$624$	0	0
$625$	−25.8026	−1.03210
$626$	10.0739	0.402634
$627$	0	0
$628$	−0.782637	−0.0312306
$629$	−15.4493	−0.616003
$630$	0	0
$631$	29.2625	1.16492	0.582461	−	0.812858i	$-0.302090\pi$
$631$	29.2625	1.16492	0.582461	+	0.812858i	$0.302090\pi$
$632$	−13.8443	−0.550695
$633$	0	0
$634$	6.31050	0.250622
$635$	37.8125	1.50054
$636$	0	0
$637$	30.1709	1.19541
$638$	−44.5381	−1.76328
$639$	0	0
$640$	2.27289	0.0898439
$641$	22.1475	0.874772	0.437386	−	0.899274i	$-0.355904\pi$
$641$	22.1475	0.874772	0.437386	+	0.899274i	$0.355904\pi$
$642$	0	0
$643$	12.5497	0.494912	0.247456	−	0.968899i	$-0.420405\pi$
$643$	12.5497	0.494912	0.247456	+	0.968899i	$0.420405\pi$
$644$	25.2990	0.996920
$645$	0	0
$646$	15.4152	0.606502
$647$	−20.7990	−0.817695	−0.408847	−	0.912603i	$-0.634069\pi$
$647$	−20.7990	−0.817695	−0.408847	+	0.912603i	$0.634069\pi$
$648$	0	0
$649$	43.4659	1.70619
$650$	0.640006	0.0251031
$651$	0	0
$652$	17.1474	0.671545
$653$	13.7151	0.536714	0.268357	−	0.963320i	$-0.413519\pi$
$653$	13.7151	0.536714	0.268357	+	0.963320i	$0.413519\pi$
$654$	0	0
$655$	46.0289	1.79850
$656$	−3.00687	−0.117398
$657$	0	0
$658$	7.19411	0.280456
$659$	−35.3140	−1.37564	−0.687819	−	0.725882i	$-0.741432\pi$
$659$	−35.3140	−1.37564	−0.687819	+	0.725882i	$0.741432\pi$
$660$	0	0
$661$	9.79629	0.381031	0.190516	−	0.981684i	$-0.438984\pi$
$661$	9.79629	0.381031	0.190516	+	0.981684i	$0.438984\pi$
$662$	−22.6231	−0.879273
$663$	0	0
$664$	1.47600	0.0572800
$665$	42.8441	1.66142
$666$	0	0
$667$	61.7549	2.39116
$668$	−8.70660	−0.336868
$669$	0	0
$670$	−32.0766	−1.23923
$671$	71.9131	2.77617
$672$	0	0
$673$	−29.9705	−1.15528	−0.577639	−	0.816293i	$-0.696026\pi$
$673$	−29.9705	−1.15528	−0.577639	+	0.816293i	$0.696026\pi$
$674$	5.87625	0.226345
$675$	0	0
$676$	1.85840	0.0714770
$677$	−43.8881	−1.68676	−0.843378	−	0.537321i	$-0.819436\pi$
$677$	−43.8881	−1.68676	−0.843378	+	0.537321i	$0.819436\pi$
$678$	0	0
$679$	−44.1609	−1.69474
$680$	7.15721	0.274466
$681$	0	0
$682$	2.33135	0.0892720
$683$	5.90641	0.226002	0.113001	−	0.993595i	$-0.463954\pi$
$683$	5.90641	0.226002	0.113001	+	0.993595i	$0.463954\pi$
$684$	0	0
$685$	−24.1004	−0.920830
$686$	3.18490	0.121600
$687$	0	0
$688$	−4.16284	−0.158707
$689$	−7.49750	−0.285632
$690$	0	0
$691$	31.1756	1.18598	0.592988	−	0.805211i	$-0.297948\pi$
$691$	31.1756	1.18598	0.592988	+	0.805211i	$0.297948\pi$
$692$	2.86163	0.108783
$693$	0	0
$694$	35.2993	1.33994
$695$	−46.0319	−1.74609
$696$	0	0
$697$	−9.46845	−0.358643
$698$	−19.4442	−0.735974
$699$	0	0
$700$	0.639332	0.0241645
$701$	−27.3130	−1.03160	−0.515798	−	0.856710i	$-0.672505\pi$
$701$	−27.3130	−1.03160	−0.515798	+	0.856710i	$0.672505\pi$
$702$	0	0
$703$	−24.0174	−0.905835
$704$	−4.73844	−0.178587
$705$	0	0
$706$	−9.38324	−0.353143
$707$	51.1357	1.92316
$708$	0	0
$709$	−47.7814	−1.79447	−0.897235	−	0.441554i	$-0.854427\pi$
$709$	−47.7814	−1.79447	−0.897235	+	0.441554i	$0.854427\pi$
$710$	25.1057	0.942199
$711$	0	0
$712$	14.8032	0.554772
$713$	−3.23256	−0.121060
$714$	0	0
$715$	−41.5145	−1.55255
$716$	−5.65183	−0.211219
$717$	0	0
$718$	1.80878	0.0675031
$719$	−17.0550	−0.636045	−0.318022	−	0.948083i	$-0.603019\pi$
$719$	−17.0550	−0.636045	−0.318022	+	0.948083i	$0.603019\pi$
$720$	0	0
$721$	36.0394	1.34218
$722$	4.96444	0.184757
$723$	0	0
$724$	−7.26999	−0.270187
$725$	1.56061	0.0579596
$726$	0	0
$727$	−7.07871	−0.262535	−0.131267	−	0.991347i	$-0.541905\pi$
$727$	−7.07871	−0.262535	−0.131267	+	0.991347i	$0.541905\pi$
$728$	14.8428	0.550109
$729$	0	0
$730$	−33.5290	−1.24096
$731$	−13.1086	−0.484838
$732$	0	0
$733$	−13.5779	−0.501510	−0.250755	−	0.968051i	$-0.580679\pi$
$733$	−13.5779	−0.501510	−0.250755	+	0.968051i	$0.580679\pi$
$734$	25.0204	0.923519
$735$	0	0
$736$	6.57014	0.242179
$737$	66.8721	2.46326
$738$	0	0
$739$	17.6807	0.650394	0.325197	−	0.945646i	$-0.394569\pi$
$739$	17.6807	0.650394	0.325197	+	0.945646i	$0.394569\pi$
$740$	−11.1512	−0.409926
$741$	0	0
$742$	−7.48960	−0.274952
$743$	−12.0012	−0.440282	−0.220141	−	0.975468i	$-0.570652\pi$
$743$	−12.0012	−0.440282	−0.220141	+	0.975468i	$0.570652\pi$
$744$	0	0
$745$	−2.27289	−0.0832723
$746$	6.60049	0.241661
$747$	0	0
$748$	−14.9211	−0.545568
$749$	43.7980	1.60034
$750$	0	0
$751$	−25.5761	−0.933284	−0.466642	−	0.884446i	$-0.654536\pi$
$751$	−25.5761	−0.933284	−0.466642	+	0.884446i	$0.654536\pi$
$752$	1.86831	0.0681302
$753$	0	0
$754$	36.2312	1.31946
$755$	21.9827	0.800032
$756$	0	0
$757$	43.2463	1.57181	0.785906	−	0.618346i	$-0.212197\pi$
$757$	43.2463	1.57181	0.785906	+	0.618346i	$0.212197\pi$
$758$	20.8003	0.755502
$759$	0	0
$760$	11.1266	0.403604
$761$	−14.6027	−0.529347	−0.264673	−	0.964338i	$-0.585264\pi$
$761$	−14.6027	−0.529347	−0.264673	+	0.964338i	$0.585264\pi$
$762$	0	0
$763$	44.7142	1.61876
$764$	−10.1345	−0.366654
$765$	0	0
$766$	14.7037	0.531265
$767$	−35.3590	−1.27674
$768$	0	0
$769$	−12.9442	−0.466780	−0.233390	−	0.972383i	$-0.574982\pi$
$769$	−12.9442	−0.466780	−0.233390	+	0.972383i	$0.574982\pi$
$770$	−41.4708	−1.49450
$771$	0	0
$772$	−18.4300	−0.663310
$773$	24.8173	0.892617	0.446309	−	0.894879i	$-0.352738\pi$
$773$	24.8173	0.892617	0.446309	+	0.894879i	$0.352738\pi$
$774$	0	0
$775$	−0.0816902	−0.00293440
$776$	−11.4686	−0.411697
$777$	0	0
$778$	−7.34324	−0.263268
$779$	−14.7197	−0.527387
$780$	0	0
$781$	−52.3394	−1.87285
$782$	20.6890	0.739837
$783$	0	0
$784$	7.82712	0.279540
$785$	−1.77885	−0.0634898
$786$	0	0
$787$	−1.47386	−0.0525374	−0.0262687	−	0.999655i	$-0.508363\pi$
$787$	−1.47386	−0.0525374	−0.0262687	+	0.999655i	$0.508363\pi$
$788$	−16.4840	−0.587218
$789$	0	0
$790$	−31.4665	−1.11953
$791$	−62.6223	−2.22659
$792$	0	0
$793$	−58.5004	−2.07741
$794$	37.4834	1.33024
$795$	0	0
$796$	−4.38822	−0.155536
$797$	−38.0752	−1.34869	−0.674346	−	0.738415i	$-0.735574\pi$
$797$	−38.0752	−1.34869	−0.674346	+	0.738415i	$0.735574\pi$
$798$	0	0
$799$	5.88320	0.208133
$800$	0.166034	0.00587020
$801$	0	0
$802$	37.9835	1.34125
$803$	69.9000	2.46672
$804$	0	0
$805$	57.5018	2.02667
$806$	−1.89652	−0.0668022
$807$	0	0
$808$	13.2799	0.467186
$809$	34.4927	1.21270	0.606349	−	0.795199i	$-0.292633\pi$
$809$	34.4927	1.21270	0.606349	+	0.795199i	$0.292633\pi$
$810$	0	0
$811$	14.8233	0.520518	0.260259	−	0.965539i	$-0.416192\pi$
$811$	14.8233	0.520518	0.260259	+	0.965539i	$0.416192\pi$
$812$	36.1930	1.27013
$813$	0	0
$814$	23.2476	0.814828
$815$	38.9742	1.36521
$816$	0	0
$817$	−20.3786	−0.712956
$818$	8.15511	0.285137
$819$	0	0
$820$	−6.83428	−0.238663
$821$	41.0333	1.43207	0.716037	−	0.698063i	$-0.245954\pi$
$821$	41.0333	1.43207	0.716037	+	0.698063i	$0.245954\pi$
$822$	0	0
$823$	−22.1509	−0.772131	−0.386065	−	0.922471i	$-0.626166\pi$
$823$	−22.1509	−0.772131	−0.386065	+	0.922471i	$0.626166\pi$
$824$	9.35943	0.326051
$825$	0	0
$826$	−35.3217	−1.22900
$827$	−44.5972	−1.55080	−0.775398	−	0.631473i	$-0.782450\pi$
$827$	−44.5972	−1.55080	−0.775398	+	0.631473i	$0.782450\pi$
$828$	0	0
$829$	17.5797	0.610567	0.305283	−	0.952262i	$-0.401249\pi$
$829$	17.5797	0.610567	0.305283	+	0.952262i	$0.401249\pi$
$830$	3.35480	0.116447
$831$	0	0
$832$	3.85466	0.133636
$833$	24.6472	0.853973
$834$	0	0
$835$	−19.7891	−0.684832
$836$	−23.1963	−0.802261
$837$	0	0
$838$	6.56704	0.226855
$839$	16.6181	0.573722	0.286861	−	0.957972i	$-0.407388\pi$
$839$	16.6181	0.573722	0.286861	+	0.957972i	$0.407388\pi$
$840$	0	0
$841$	59.3472	2.04646
$842$	−2.38560	−0.0822133
$843$	0	0
$844$	−9.44934	−0.325260
$845$	4.22394	0.145308
$846$	0	0
$847$	44.1001	1.51530
$848$	−1.94505	−0.0667932
$849$	0	0
$850$	0.522833	0.0179330
$851$	−32.2343	−1.10498
$852$	0	0
$853$	23.4444	0.802720	0.401360	−	0.915920i	$-0.368538\pi$
$853$	23.4444	0.802720	0.401360	+	0.915920i	$0.368538\pi$
$854$	−58.4388	−1.99973
$855$	0	0
$856$	11.3743	0.388766
$857$	−44.9801	−1.53649	−0.768246	−	0.640154i	$-0.778870\pi$
$857$	−44.9801	−1.53649	−0.768246	+	0.640154i	$0.778870\pi$
$858$	0	0
$859$	31.3441	1.06945	0.534723	−	0.845027i	$-0.320416\pi$
$859$	31.3441	1.06945	0.534723	+	0.845027i	$0.320416\pi$
$860$	−9.46169	−0.322641
$861$	0	0
$862$	5.91578	0.201492
$863$	31.1561	1.06057	0.530283	−	0.847821i	$-0.322086\pi$
$863$	31.1561	1.06057	0.530283	+	0.847821i	$0.322086\pi$
$864$	0	0
$865$	6.50418	0.221149
$866$	−18.6739	−0.634566
$867$	0	0
$868$	−1.89453	−0.0643044
$869$	65.6001	2.22533
$870$	0	0
$871$	−54.3996	−1.84326
$872$	11.6123	0.393241
$873$	0	0
$874$	32.1631	1.08793
$875$	−42.3068	−1.43023
$876$	0	0
$877$	−32.3808	−1.09342	−0.546711	−	0.837322i	$-0.684120\pi$
$877$	−32.3808	−1.09342	−0.546711	+	0.837322i	$0.684120\pi$
$878$	−36.4850	−1.23131
$879$	0	0
$880$	−10.7700	−0.363055
$881$	−15.8533	−0.534113	−0.267056	−	0.963681i	$-0.586051\pi$
$881$	−15.8533	−0.534113	−0.267056	+	0.963681i	$0.586051\pi$
$882$	0	0
$883$	−17.1753	−0.577994	−0.288997	−	0.957330i	$-0.593322\pi$
$883$	−17.1753	−0.577994	−0.288997	+	0.957330i	$0.593322\pi$
$884$	12.1381	0.408249
$885$	0	0
$886$	0.771418	0.0259163
$887$	0.759463	0.0255003	0.0127501	−	0.999919i	$-0.495941\pi$
$887$	0.759463	0.0255003	0.0127501	+	0.999919i	$0.495941\pi$
$888$	0	0
$889$	64.0597	2.14849
$890$	33.6460	1.12782
$891$	0	0
$892$	−5.39202	−0.180538
$893$	9.14602	0.306060
$894$	0	0
$895$	−12.8460	−0.429394
$896$	3.85060	0.128640
$897$	0	0
$898$	4.71625	0.157383
$899$	−4.62454	−0.154237
$900$	0	0
$901$	−6.12485	−0.204048
$902$	14.2479	0.474402
$903$	0	0
$904$	−16.2630	−0.540899
$905$	−16.5239	−0.549273
$906$	0	0
$907$	13.5094	0.448573	0.224287	−	0.974523i	$-0.427995\pi$
$907$	13.5094	0.448573	0.224287	+	0.974523i	$0.427995\pi$
$908$	25.4926	0.846003
$909$	0	0
$910$	33.7360	1.11834
$911$	−37.4589	−1.24107	−0.620535	−	0.784179i	$-0.713085\pi$
$911$	−37.4589	−1.24107	−0.620535	+	0.784179i	$0.713085\pi$
$912$	0	0
$913$	−6.99395	−0.231466
$914$	−39.2811	−1.29930
$915$	0	0
$916$	23.7053	0.783246
$917$	77.9795	2.57511
$918$	0	0
$919$	−37.0300	−1.22151	−0.610753	−	0.791821i	$-0.709133\pi$
$919$	−37.0300	−1.22151	−0.610753	+	0.791821i	$0.709133\pi$
$920$	14.9332	0.492334
$921$	0	0
$922$	−16.5258	−0.544249
$923$	42.5774	1.40145
$924$	0	0
$925$	−0.814594	−0.0267837
$926$	17.3522	0.570229
$927$	0	0
$928$	9.39932	0.308548
$929$	55.1715	1.81012	0.905059	−	0.425285i	$-0.139826\pi$
$929$	55.1715	1.81012	0.905059	+	0.425285i	$0.139826\pi$
$930$	0	0
$931$	38.3165	1.25577
$932$	−22.3704	−0.732767
$933$	0	0
$934$	12.8545	0.420611
$935$	−33.9140	−1.10911
$936$	0	0
$937$	48.0238	1.56887	0.784435	−	0.620211i	$-0.212953\pi$
$937$	48.0238	1.56887	0.784435	+	0.620211i	$0.212953\pi$
$938$	−54.3423	−1.77434
$939$	0	0
$940$	4.24646	0.138504
$941$	−21.1915	−0.690824	−0.345412	−	0.938451i	$-0.612261\pi$
$941$	−21.1915	−0.690824	−0.345412	+	0.938451i	$0.612261\pi$
$942$	0	0
$943$	−19.7555	−0.643329
$944$	−9.17305	−0.298557
$945$	0	0
$946$	19.7254	0.641327
$947$	−5.74946	−0.186832	−0.0934161	−	0.995627i	$-0.529779\pi$
$947$	−5.74946	−0.186832	−0.0934161	+	0.995627i	$0.529779\pi$
$948$	0	0
$949$	−56.8628	−1.84584
$950$	0.812796	0.0263706
$951$	0	0
$952$	12.1253	0.392984
$953$	−24.5106	−0.793975	−0.396988	−	0.917824i	$-0.629944\pi$
$953$	−24.5106	−0.793975	−0.396988	+	0.917824i	$0.629944\pi$
$954$	0	0
$955$	−23.0347	−0.745384
$956$	0.194267	0.00628304
$957$	0	0
$958$	20.4482	0.660652
$959$	−40.8295	−1.31845
$960$	0	0
$961$	−30.7579	−0.992191
$962$	−18.9116	−0.609736
$963$	0	0
$964$	24.1769	0.778685
$965$	−41.8893	−1.34847
$966$	0	0
$967$	−62.0615	−1.99576	−0.997881	−	0.0650586i	$-0.979277\pi$
$967$	−62.0615	−1.99576	−0.997881	+	0.0650586i	$0.979277\pi$
$968$	11.4528	0.368107
$969$	0	0
$970$	−26.0668	−0.836954
$971$	6.47793	0.207887	0.103943	−	0.994583i	$-0.466854\pi$
$971$	6.47793	0.207887	0.103943	+	0.994583i	$0.466854\pi$
$972$	0	0
$973$	−77.9845	−2.50007
$974$	−20.3997	−0.653649
$975$	0	0
$976$	−15.1765	−0.485789
$977$	30.0328	0.960833	0.480416	−	0.877040i	$-0.340486\pi$
$977$	30.0328	0.960833	0.480416	+	0.877040i	$0.340486\pi$
$978$	0	0
$979$	−70.1439	−2.24181
$980$	17.7902	0.568287
$981$	0	0
$982$	−17.5220	−0.559149
$983$	−29.6421	−0.945437	−0.472719	−	0.881213i	$-0.656727\pi$
$983$	−29.6421	−0.945437	−0.472719	+	0.881213i	$0.656727\pi$
$984$	0	0
$985$	−37.4663	−1.19378
$986$	29.5979	0.942590
$987$	0	0
$988$	18.8699	0.600332
$989$	−27.3505	−0.869695
$990$	0	0
$991$	33.8924	1.07663	0.538314	−	0.842744i	$-0.319061\pi$
$991$	33.8924	1.07663	0.538314	+	0.842744i	$0.319061\pi$
$992$	−0.492008	−0.0156213
$993$	0	0
$994$	42.5326	1.34905
$995$	−9.97395	−0.316196
$996$	0	0
$997$	−44.7190	−1.41627	−0.708133	−	0.706079i	$-0.750462\pi$
$997$	−44.7190	−1.41627	−0.708133	+	0.706079i	$0.750462\pi$
$998$	−23.2421	−0.735717
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.p.1.9	yes	12
3.2	odd	2		8046.2.a.i.1.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.i.1.4	✓	12	3.2	odd	2
8046.2.a.p.1.9	yes	12	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.27289 q^{5} +3.85060 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.27289 q^{5} +3.85060 q^{7} +1.00000 q^{8} +2.27289 q^{10} -4.73844 q^{11} +3.85466 q^{13} +3.85060 q^{14} +1.00000 q^{16} +3.14894 q^{17} +4.89535 q^{19} +2.27289 q^{20} -4.73844 q^{22} +6.57014 q^{23} +0.166034 q^{25} +3.85466 q^{26} +3.85060 q^{28} +9.39932 q^{29} -0.492008 q^{31} +1.00000 q^{32} +3.14894 q^{34} +8.75199 q^{35} -4.90618 q^{37} +4.89535 q^{38} +2.27289 q^{40} -3.00687 q^{41} -4.16284 q^{43} -4.73844 q^{44} +6.57014 q^{46} +1.86831 q^{47} +7.82712 q^{49} +0.166034 q^{50} +3.85466 q^{52} -1.94505 q^{53} -10.7700 q^{55} +3.85060 q^{56} +9.39932 q^{58} -9.17305 q^{59} -15.1765 q^{61} -0.492008 q^{62} +1.00000 q^{64} +8.76122 q^{65} -14.1127 q^{67} +3.14894 q^{68} +8.75199 q^{70} +11.0457 q^{71} -14.7517 q^{73} -4.90618 q^{74} +4.89535 q^{76} -18.2458 q^{77} -13.8443 q^{79} +2.27289 q^{80} -3.00687 q^{82} +1.47600 q^{83} +7.15721 q^{85} -4.16284 q^{86} -4.73844 q^{88} +14.8032 q^{89} +14.8428 q^{91} +6.57014 q^{92} +1.86831 q^{94} +11.1266 q^{95} -11.4686 q^{97} +7.82712 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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