LMFDB - Embedded newform 8001.2.a.n.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
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} - 3 x^{10} + 41 x^{9} - 11 x^{8} - 123 x^{7} + 44 x^{6} + 159 x^{5} - 39 x^{4} + \cdots - 1 $ x^12 - 5x^11 - 3x^10 + 41x^9 - 11x^8 - 123x^7 + 44x^6 + 159x^5 - 39x^4 - 71x^3 + 16x^2 + 7*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.920581$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.92058 q^{2} +1.68863 q^{4} -0.753423 q^{5} +1.00000 q^{7} -0.598010 q^{8} +O(q^{10})$	\(q+1.92058 q^{2} +1.68863 q^{4} -0.753423 q^{5} +1.00000 q^{7} -0.598010 q^{8} -1.44701 q^{10} +3.22310 q^{11} -4.98449 q^{13} +1.92058 q^{14} -4.52579 q^{16} +6.32104 q^{17} -2.95721 q^{19} -1.27225 q^{20} +6.19022 q^{22} -0.477419 q^{23} -4.43235 q^{25} -9.57312 q^{26} +1.68863 q^{28} +6.23114 q^{29} +9.65207 q^{31} -7.49612 q^{32} +12.1401 q^{34} -0.753423 q^{35} -0.00252595 q^{37} -5.67957 q^{38} +0.450555 q^{40} +7.44774 q^{41} -0.802098 q^{43} +5.44262 q^{44} -0.916921 q^{46} -7.63168 q^{47} +1.00000 q^{49} -8.51269 q^{50} -8.41696 q^{52} -1.03403 q^{53} -2.42836 q^{55} -0.598010 q^{56} +11.9674 q^{58} -1.36149 q^{59} +10.9451 q^{61} +18.5376 q^{62} -5.34533 q^{64} +3.75543 q^{65} +12.1648 q^{67} +10.6739 q^{68} -1.44701 q^{70} +9.50724 q^{71} -2.92872 q^{73} -0.00485128 q^{74} -4.99364 q^{76} +3.22310 q^{77} +14.7112 q^{79} +3.40983 q^{80} +14.3040 q^{82} +12.3257 q^{83} -4.76242 q^{85} -1.54049 q^{86} -1.92744 q^{88} -2.65289 q^{89} -4.98449 q^{91} -0.806184 q^{92} -14.6573 q^{94} +2.22803 q^{95} +9.13964 q^{97} +1.92058 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * 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 $

$2$	1.92058	1.35806	0.679028	−	0.734112i	$-0.262402\pi$
$2$	1.92058	1.35806	0.679028	+	0.734112i	$0.262402\pi$
$3$	0	0
$3$	0	0
$4$	1.68863	0.844315
$5$	−0.753423	−0.336941	−0.168471	−	0.985707i	$-0.553883\pi$
$5$	−0.753423	−0.336941	−0.168471	+	0.985707i	$0.553883\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.598010	−0.211428
$9$	0	0
$10$	−1.44701	−0.457585
$11$	3.22310	0.971800	0.485900	−	0.874014i	$-0.338492\pi$
$11$	3.22310	0.971800	0.485900	+	0.874014i	$0.338492\pi$
$12$	0	0
$13$	−4.98449	−1.38245	−0.691224	−	0.722640i	$-0.742928\pi$
$13$	−4.98449	−1.38245	−0.691224	+	0.722640i	$0.742928\pi$
$14$	1.92058	0.513297
$15$	0	0
$16$	−4.52579	−1.13145
$17$	6.32104	1.53308	0.766539	−	0.642198i	$-0.221977\pi$
$17$	6.32104	1.53308	0.766539	+	0.642198i	$0.221977\pi$
$18$	0	0
$19$	−2.95721	−0.678431	−0.339216	−	0.940709i	$-0.610162\pi$
$19$	−2.95721	−0.678431	−0.339216	+	0.940709i	$0.610162\pi$
$20$	−1.27225	−0.284485
$21$	0	0
$22$	6.19022	1.31976
$23$	−0.477419	−0.0995487	−0.0497743	−	0.998760i	$-0.515850\pi$
$23$	−0.477419	−0.0995487	−0.0497743	+	0.998760i	$0.515850\pi$
$24$	0	0
$25$	−4.43235	−0.886471
$26$	−9.57312	−1.87744
$27$	0	0
$28$	1.68863	0.319121
$29$	6.23114	1.15709	0.578547	−	0.815649i	$-0.303620\pi$
$29$	6.23114	1.15709	0.578547	+	0.815649i	$0.303620\pi$
$30$	0	0
$31$	9.65207	1.73356	0.866781	−	0.498688i	$-0.166185\pi$
$31$	9.65207	1.73356	0.866781	+	0.498688i	$0.166185\pi$
$32$	−7.49612	−1.32514
$33$	0	0
$34$	12.1401	2.08200
$35$	−0.753423	−0.127352
$36$	0	0
$37$	−0.00252595	−0.000415263 0	−0.000207632	−	1.00000i	$-0.500066\pi$
$37$	−0.00252595	−0.000415263 0	−0.000207632		1.00000i	$0.500066\pi$
$38$	−5.67957	−0.921348
$39$	0	0
$40$	0.450555	0.0712389
$41$	7.44774	1.16314	0.581571	−	0.813496i	$-0.302438\pi$
$41$	7.44774	1.16314	0.581571	+	0.813496i	$0.302438\pi$
$42$	0	0
$43$	−0.802098	−0.122319	−0.0611594	−	0.998128i	$-0.519480\pi$
$43$	−0.802098	−0.122319	−0.0611594	+	0.998128i	$0.519480\pi$
$44$	5.44262	0.820506
$45$	0	0
$46$	−0.916921	−0.135193
$47$	−7.63168	−1.11319	−0.556597	−	0.830782i	$-0.687893\pi$
$47$	−7.63168	−1.11319	−0.556597	+	0.830782i	$0.687893\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−8.51269	−1.20388
$51$	0	0
$52$	−8.41696	−1.16722
$53$	−1.03403	−0.142035	−0.0710174	−	0.997475i	$-0.522625\pi$
$53$	−1.03403	−0.142035	−0.0710174	+	0.997475i	$0.522625\pi$
$54$	0	0
$55$	−2.42836	−0.327440
$56$	−0.598010	−0.0799124
$57$	0	0
$58$	11.9674	1.57140
$59$	−1.36149	−0.177251	−0.0886253	−	0.996065i	$-0.528247\pi$
$59$	−1.36149	−0.177251	−0.0886253	+	0.996065i	$0.528247\pi$
$60$	0	0
$61$	10.9451	1.40138	0.700690	−	0.713466i	$-0.252876\pi$
$61$	10.9451	1.40138	0.700690	+	0.713466i	$0.252876\pi$
$62$	18.5376	2.35428
$63$	0	0
$64$	−5.34533	−0.668166
$65$	3.75543	0.465804
$66$	0	0
$67$	12.1648	1.48617	0.743084	−	0.669198i	$-0.233362\pi$
$67$	12.1648	1.48617	0.743084	+	0.669198i	$0.233362\pi$
$68$	10.6739	1.29440
$69$	0	0
$70$	−1.44701	−0.172951
$71$	9.50724	1.12830	0.564151	−	0.825672i	$-0.309204\pi$
$71$	9.50724	1.12830	0.564151	+	0.825672i	$0.309204\pi$
$72$	0	0
$73$	−2.92872	−0.342781	−0.171390	−	0.985203i	$-0.554826\pi$
$73$	−2.92872	−0.342781	−0.171390	+	0.985203i	$0.554826\pi$
$74$	−0.00485128	−0.000563950 0
$75$	0	0
$76$	−4.99364	−0.572810
$77$	3.22310	0.367306
$78$	0	0
$79$	14.7112	1.65514	0.827572	−	0.561359i	$-0.189721\pi$
$79$	14.7112	1.65514	0.827572	+	0.561359i	$0.189721\pi$
$80$	3.40983	0.381231
$81$	0	0
$82$	14.3040	1.57961
$83$	12.3257	1.35292	0.676460	−	0.736479i	$-0.263513\pi$
$83$	12.3257	1.35292	0.676460	+	0.736479i	$0.263513\pi$
$84$	0	0
$85$	−4.76242	−0.516557
$86$	−1.54049	−0.166116
$87$	0	0
$88$	−1.92744	−0.205466
$89$	−2.65289	−0.281205	−0.140603	−	0.990066i	$-0.544904\pi$
$89$	−2.65289	−0.281205	−0.140603	+	0.990066i	$0.544904\pi$
$90$	0	0
$91$	−4.98449	−0.522517
$92$	−0.806184	−0.0840505
$93$	0	0
$94$	−14.6573	−1.51178
$95$	2.22803	0.228591
$96$	0	0
$97$	9.13964	0.927990	0.463995	−	0.885838i	$-0.346416\pi$
$97$	9.13964	0.927990	0.463995	+	0.885838i	$0.346416\pi$
$98$	1.92058	0.194008
$99$	0	0
$100$	−7.48461	−0.748461
$101$	−7.26444	−0.722839	−0.361419	−	0.932403i	$-0.617708\pi$
$101$	−7.26444	−0.722839	−0.361419	+	0.932403i	$0.617708\pi$
$102$	0	0
$103$	−17.8783	−1.76160	−0.880802	−	0.473485i	$-0.842996\pi$
$103$	−17.8783	−1.76160	−0.880802	+	0.473485i	$0.842996\pi$
$104$	2.98077	0.292289
$105$	0	0
$106$	−1.98594	−0.192891
$107$	−1.42488	−0.137748	−0.0688740	−	0.997625i	$-0.521941\pi$
$107$	−1.42488	−0.137748	−0.0688740	+	0.997625i	$0.521941\pi$
$108$	0	0
$109$	−12.3853	−1.18629	−0.593147	−	0.805094i	$-0.702115\pi$
$109$	−12.3853	−1.18629	−0.593147	+	0.805094i	$0.702115\pi$
$110$	−4.66385	−0.444681
$111$	0	0
$112$	−4.52579	−0.427647
$113$	15.4026	1.44895	0.724475	−	0.689301i	$-0.242082\pi$
$113$	15.4026	1.44895	0.724475	+	0.689301i	$0.242082\pi$
$114$	0	0
$115$	0.359698	0.0335420
$116$	10.5221	0.976953
$117$	0	0
$118$	−2.61485	−0.240716
$119$	6.32104	0.579449
$120$	0	0
$121$	−0.611645	−0.0556041
$122$	21.0210	1.90315
$123$	0	0
$124$	16.2988	1.46367
$125$	7.10655	0.635630
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	4.72610	0.417732
$129$	0	0
$130$	7.21261	0.632588
$131$	−12.0709	−1.05464	−0.527319	−	0.849667i	$-0.676803\pi$
$131$	−12.0709	−1.05464	−0.527319	+	0.849667i	$0.676803\pi$
$132$	0	0
$133$	−2.95721	−0.256423
$134$	23.3635	2.01830
$135$	0	0
$136$	−3.78004	−0.324136
$137$	−1.11071	−0.0948945	−0.0474473	−	0.998874i	$-0.515109\pi$
$137$	−1.11071	−0.0948945	−0.0474473	+	0.998874i	$0.515109\pi$
$138$	0	0
$139$	−18.9836	−1.61016	−0.805082	−	0.593163i	$-0.797879\pi$
$139$	−18.9836	−1.61016	−0.805082	+	0.593163i	$0.797879\pi$
$140$	−1.27225	−0.107525
$141$	0	0
$142$	18.2594	1.53230
$143$	−16.0655	−1.34346
$144$	0	0
$145$	−4.69469	−0.389873
$146$	−5.62484	−0.465515
$147$	0	0
$148$	−0.00426539	−0.000350613 0
$149$	6.14569	0.503474	0.251737	−	0.967796i	$-0.418998\pi$
$149$	6.14569	0.503474	0.251737	+	0.967796i	$0.418998\pi$
$150$	0	0
$151$	17.9024	1.45688	0.728438	−	0.685112i	$-0.240247\pi$
$151$	17.9024	1.45688	0.728438	+	0.685112i	$0.240247\pi$
$152$	1.76844	0.143440
$153$	0	0
$154$	6.19022	0.498822
$155$	−7.27209	−0.584109
$156$	0	0
$157$	5.21149	0.415922	0.207961	−	0.978137i	$-0.433317\pi$
$157$	5.21149	0.415922	0.207961	+	0.978137i	$0.433317\pi$
$158$	28.2541	2.24778
$159$	0	0
$160$	5.64775	0.446494
$161$	−0.477419	−0.0376259
$162$	0	0
$163$	17.5364	1.37355	0.686777	−	0.726868i	$-0.259025\pi$
$163$	17.5364	1.37355	0.686777	+	0.726868i	$0.259025\pi$
$164$	12.5765	0.982058
$165$	0	0
$166$	23.6725	1.83734
$167$	20.7361	1.60460	0.802302	−	0.596918i	$-0.203608\pi$
$167$	20.7361	1.60460	0.802302	+	0.596918i	$0.203608\pi$
$168$	0	0
$169$	11.8451	0.911165
$170$	−9.14661	−0.701513
$171$	0	0
$172$	−1.35445	−0.103276
$173$	7.35863	0.559466	0.279733	−	0.960078i	$-0.409754\pi$
$173$	7.35863	0.559466	0.279733	+	0.960078i	$0.409754\pi$
$174$	0	0
$175$	−4.43235	−0.335054
$176$	−14.5871	−1.09954
$177$	0	0
$178$	−5.09508	−0.381893
$179$	8.62009	0.644296	0.322148	−	0.946689i	$-0.395595\pi$
$179$	8.62009	0.644296	0.322148	+	0.946689i	$0.395595\pi$
$180$	0	0
$181$	16.9545	1.26022	0.630109	−	0.776507i	$-0.283010\pi$
$181$	16.9545	1.26022	0.630109	+	0.776507i	$0.283010\pi$
$182$	−9.57312	−0.709607
$183$	0	0
$184$	0.285501	0.0210474
$185$	0.00190311	0.000139919 0
$186$	0	0
$187$	20.3733	1.48985
$188$	−12.8871	−0.939887
$189$	0	0
$190$	4.27912	0.310440
$191$	15.6440	1.13196	0.565980	−	0.824419i	$-0.308498\pi$
$191$	15.6440	1.13196	0.565980	+	0.824419i	$0.308498\pi$
$192$	0	0
$193$	−4.36633	−0.314295	−0.157148	−	0.987575i	$-0.550230\pi$
$193$	−4.36633	−0.314295	−0.157148	+	0.987575i	$0.550230\pi$
$194$	17.5534	1.26026
$195$	0	0
$196$	1.68863	0.120616
$197$	−9.41530	−0.670812	−0.335406	−	0.942074i	$-0.608873\pi$
$197$	−9.41530	−0.670812	−0.335406	+	0.942074i	$0.608873\pi$
$198$	0	0
$199$	−20.8307	−1.47665	−0.738326	−	0.674444i	$-0.764384\pi$
$199$	−20.8307	−1.47665	−0.738326	+	0.674444i	$0.764384\pi$
$200$	2.65059	0.187425
$201$	0	0
$202$	−13.9519	−0.981655
$203$	6.23114	0.437341
$204$	0	0
$205$	−5.61130	−0.391910
$206$	−34.3368	−2.39236
$207$	0	0
$208$	22.5587	1.56417
$209$	−9.53139	−0.659300
$210$	0	0
$211$	24.2316	1.66817	0.834085	−	0.551635i	$-0.185996\pi$
$211$	24.2316	1.66817	0.834085	+	0.551635i	$0.185996\pi$
$212$	−1.74609	−0.119922
$213$	0	0
$214$	−2.73659	−0.187070
$215$	0.604319	0.0412142
$216$	0	0
$217$	9.65207	0.655225
$218$	−23.7869	−1.61105
$219$	0	0
$220$	−4.10060	−0.276462
$221$	−31.5072	−2.11940
$222$	0	0
$223$	4.29516	0.287625	0.143812	−	0.989605i	$-0.454064\pi$
$223$	4.29516	0.287625	0.143812	+	0.989605i	$0.454064\pi$
$224$	−7.49612	−0.500856
$225$	0	0
$226$	29.5819	1.96776
$227$	−25.6070	−1.69959	−0.849797	−	0.527110i	$-0.823276\pi$
$227$	−25.6070	−1.69959	−0.849797	+	0.527110i	$0.823276\pi$
$228$	0	0
$229$	19.5476	1.29174	0.645871	−	0.763447i	$-0.276494\pi$
$229$	19.5476	1.29174	0.645871	+	0.763447i	$0.276494\pi$
$230$	0.690830	0.0455520
$231$	0	0
$232$	−3.72629	−0.244643
$233$	−5.46869	−0.358265	−0.179133	−	0.983825i	$-0.557329\pi$
$233$	−5.46869	−0.358265	−0.179133	+	0.983825i	$0.557329\pi$
$234$	0	0
$235$	5.74988	0.375081
$236$	−2.29905	−0.149655
$237$	0	0
$238$	12.1401	0.786924
$239$	−9.53365	−0.616681	−0.308340	−	0.951276i	$-0.599774\pi$
$239$	−9.53365	−0.616681	−0.308340	+	0.951276i	$0.599774\pi$
$240$	0	0
$241$	−23.2075	−1.49493	−0.747463	−	0.664303i	$-0.768728\pi$
$241$	−23.2075	−1.49493	−0.747463	+	0.664303i	$0.768728\pi$
$242$	−1.17471	−0.0755135
$243$	0	0
$244$	18.4823	1.18321
$245$	−0.753423	−0.0481344
$246$	0	0
$247$	14.7402	0.937897
$248$	−5.77203	−0.366524
$249$	0	0
$250$	13.6487	0.863220
$251$	−0.624486	−0.0394172	−0.0197086	−	0.999806i	$-0.506274\pi$
$251$	−0.624486	−0.0394172	−0.0197086	+	0.999806i	$0.506274\pi$
$252$	0	0
$253$	−1.53877	−0.0967414
$254$	1.92058	0.120508
$255$	0	0
$256$	19.7675	1.23547
$257$	−2.87827	−0.179541	−0.0897707	−	0.995962i	$-0.528613\pi$
$257$	−2.87827	−0.179541	−0.0897707	+	0.995962i	$0.528613\pi$
$258$	0	0
$259$	−0.00252595	−0.000156955 0
$260$	6.34154	0.393285
$261$	0	0
$262$	−23.1831	−1.43226
$263$	−7.27046	−0.448316	−0.224158	−	0.974553i	$-0.571963\pi$
$263$	−7.27046	−0.448316	−0.224158	+	0.974553i	$0.571963\pi$
$264$	0	0
$265$	0.779062	0.0478574
$266$	−5.67957	−0.348237
$267$	0	0
$268$	20.5419	1.25479
$269$	13.2286	0.806563	0.403282	−	0.915076i	$-0.367869\pi$
$269$	13.2286	0.806563	0.403282	+	0.915076i	$0.367869\pi$
$270$	0	0
$271$	−31.0471	−1.88598	−0.942988	−	0.332828i	$-0.891997\pi$
$271$	−31.0471	−1.88598	−0.942988	+	0.332828i	$0.891997\pi$
$272$	−28.6077	−1.73460
$273$	0	0
$274$	−2.13321	−0.128872
$275$	−14.2859	−0.861472
$276$	0	0
$277$	1.73394	0.104182	0.0520912	−	0.998642i	$-0.483411\pi$
$277$	1.73394	0.104182	0.0520912	+	0.998642i	$0.483411\pi$
$278$	−36.4595	−2.18669
$279$	0	0
$280$	0.450555	0.0269258
$281$	27.9088	1.66490	0.832449	−	0.554101i	$-0.186938\pi$
$281$	27.9088	1.66490	0.832449	+	0.554101i	$0.186938\pi$
$282$	0	0
$283$	−9.59401	−0.570305	−0.285152	−	0.958482i	$-0.592044\pi$
$283$	−9.59401	−0.570305	−0.285152	+	0.958482i	$0.592044\pi$
$284$	16.0542	0.952642
$285$	0	0
$286$	−30.8551	−1.82450
$287$	7.44774	0.439626
$288$	0	0
$289$	22.9555	1.35033
$290$	−9.01653	−0.529469
$291$	0	0
$292$	−4.94553	−0.289415
$293$	7.48372	0.437203	0.218602	−	0.975814i	$-0.429850\pi$
$293$	7.48372	0.437203	0.218602	+	0.975814i	$0.429850\pi$
$294$	0	0
$295$	1.02578	0.0597230
$296$	0.00151054	8.77984e−5 0
$297$	0	0
$298$	11.8033	0.683746
$299$	2.37969	0.137621
$300$	0	0
$301$	−0.802098	−0.0462322
$302$	34.3830	1.97852
$303$	0	0
$304$	13.3837	0.767609
$305$	−8.24631	−0.472182
$306$	0	0
$307$	−2.38431	−0.136080	−0.0680400	−	0.997683i	$-0.521675\pi$
$307$	−2.38431	−0.136080	−0.0680400	+	0.997683i	$0.521675\pi$
$308$	5.44262	0.310122
$309$	0	0
$310$	−13.9666	−0.793252
$311$	12.3250	0.698886	0.349443	−	0.936958i	$-0.386371\pi$
$311$	12.3250	0.698886	0.349443	+	0.936958i	$0.386371\pi$
$312$	0	0
$313$	−17.3574	−0.981099	−0.490549	−	0.871413i	$-0.663204\pi$
$313$	−17.3574	−0.981099	−0.490549	+	0.871413i	$0.663204\pi$
$314$	10.0091	0.564846
$315$	0	0
$316$	24.8419	1.39746
$317$	8.53606	0.479433	0.239716	−	0.970843i	$-0.422946\pi$
$317$	8.53606	0.479433	0.239716	+	0.970843i	$0.422946\pi$
$318$	0	0
$319$	20.0836	1.12446
$320$	4.02730	0.225133
$321$	0	0
$322$	−0.916921	−0.0510980
$323$	−18.6927	−1.04009
$324$	0	0
$325$	22.0930	1.22550
$326$	33.6800	1.86536
$327$	0	0
$328$	−4.45382	−0.245921
$329$	−7.63168	−0.420748
$330$	0	0
$331$	−3.33551	−0.183336	−0.0916681	−	0.995790i	$-0.529220\pi$
$331$	−3.33551	−0.183336	−0.0916681	+	0.995790i	$0.529220\pi$
$332$	20.8135	1.14229
$333$	0	0
$334$	39.8253	2.17914
$335$	−9.16525	−0.500751
$336$	0	0
$337$	23.4213	1.27584	0.637921	−	0.770102i	$-0.279795\pi$
$337$	23.4213	1.27584	0.637921	+	0.770102i	$0.279795\pi$
$338$	22.7496	1.23741
$339$	0	0
$340$	−8.04197	−0.436137
$341$	31.1096	1.68468
$342$	0	0
$343$	1.00000	0.0539949
$344$	0.479663	0.0258617
$345$	0	0
$346$	14.1328	0.759786
$347$	22.3771	1.20127	0.600634	−	0.799524i	$-0.294915\pi$
$347$	22.3771	1.20127	0.600634	+	0.799524i	$0.294915\pi$
$348$	0	0
$349$	−1.46642	−0.0784956	−0.0392478	−	0.999230i	$-0.512496\pi$
$349$	−1.46642	−0.0784956	−0.0392478	+	0.999230i	$0.512496\pi$
$350$	−8.51269	−0.455023
$351$	0	0
$352$	−24.1607	−1.28777
$353$	−10.0977	−0.537447	−0.268724	−	0.963217i	$-0.586602\pi$
$353$	−10.0977	−0.537447	−0.268724	+	0.963217i	$0.586602\pi$
$354$	0	0
$355$	−7.16297	−0.380171
$356$	−4.47975	−0.237426
$357$	0	0
$358$	16.5556	0.874990
$359$	34.5192	1.82185	0.910927	−	0.412567i	$-0.135368\pi$
$359$	34.5192	1.82185	0.910927	+	0.412567i	$0.135368\pi$
$360$	0	0
$361$	−10.2549	−0.539731
$362$	32.5625	1.71145
$363$	0	0
$364$	−8.41696	−0.441169
$365$	2.20657	0.115497
$366$	0	0
$367$	18.6620	0.974148	0.487074	−	0.873361i	$-0.338064\pi$
$367$	18.6620	0.974148	0.487074	+	0.873361i	$0.338064\pi$
$368$	2.16070	0.112634
$369$	0	0
$370$	0.00365507	0.000190018 0
$371$	−1.03403	−0.0536841
$372$	0	0
$373$	−27.2684	−1.41190	−0.705952	−	0.708260i	$-0.749481\pi$
$373$	−27.2684	−1.41190	−0.705952	+	0.708260i	$0.749481\pi$
$374$	39.1286	2.02329
$375$	0	0
$376$	4.56382	0.235361
$377$	−31.0591	−1.59962
$378$	0	0
$379$	−33.4354	−1.71746	−0.858731	−	0.512427i	$-0.828747\pi$
$379$	−33.4354	−1.71746	−0.858731	+	0.512427i	$0.828747\pi$
$380$	3.76233	0.193003
$381$	0	0
$382$	30.0456	1.53726
$383$	−16.7980	−0.858336	−0.429168	−	0.903225i	$-0.641193\pi$
$383$	−16.7980	−0.858336	−0.429168	+	0.903225i	$0.641193\pi$
$384$	0	0
$385$	−2.42836	−0.123761
$386$	−8.38588	−0.426830
$387$	0	0
$388$	15.4335	0.783516
$389$	−15.0006	−0.760559	−0.380280	−	0.924872i	$-0.624172\pi$
$389$	−15.0006	−0.760559	−0.380280	+	0.924872i	$0.624172\pi$
$390$	0	0
$391$	−3.01778	−0.152616
$392$	−0.598010	−0.0302041
$393$	0	0
$394$	−18.0828	−0.911000
$395$	−11.0838	−0.557686
$396$	0	0
$397$	−2.85927	−0.143503	−0.0717513	−	0.997423i	$-0.522859\pi$
$397$	−2.85927	−0.143503	−0.0717513	+	0.997423i	$0.522859\pi$
$398$	−40.0071	−2.00538
$399$	0	0
$400$	20.0599	1.00299
$401$	−0.765457	−0.0382251	−0.0191125	−	0.999817i	$-0.506084\pi$
$401$	−0.765457	−0.0382251	−0.0191125	+	0.999817i	$0.506084\pi$
$402$	0	0
$403$	−48.1107	−2.39656
$404$	−12.2670	−0.610304
$405$	0	0
$406$	11.9674	0.593933
$407$	−0.00814137	−0.000403553 0
$408$	0	0
$409$	−1.51904	−0.0751117	−0.0375559	−	0.999295i	$-0.511957\pi$
$409$	−1.51904	−0.0751117	−0.0375559	+	0.999295i	$0.511957\pi$
$410$	−10.7770	−0.532236
$411$	0	0
$412$	−30.1899	−1.48735
$413$	−1.36149	−0.0669944
$414$	0	0
$415$	−9.28647	−0.455855
$416$	37.3643	1.83194
$417$	0	0
$418$	−18.3058	−0.895366
$419$	−2.99002	−0.146072	−0.0730360	−	0.997329i	$-0.523269\pi$
$419$	−2.99002	−0.146072	−0.0730360	+	0.997329i	$0.523269\pi$
$420$	0	0
$421$	−21.5445	−1.05002	−0.525008	−	0.851097i	$-0.675938\pi$
$421$	−21.5445	−1.05002	−0.525008	+	0.851097i	$0.675938\pi$
$422$	46.5387	2.26547
$423$	0	0
$424$	0.618360	0.0300302
$425$	−28.0171	−1.35903
$426$	0	0
$427$	10.9451	0.529672
$428$	−2.40609	−0.116303
$429$	0	0
$430$	1.16064	0.0559712
$431$	18.1458	0.874055	0.437027	−	0.899448i	$-0.356031\pi$
$431$	18.1458	0.874055	0.437027	+	0.899448i	$0.356031\pi$
$432$	0	0
$433$	30.9949	1.48952	0.744759	−	0.667334i	$-0.232564\pi$
$433$	30.9949	1.48952	0.744759	+	0.667334i	$0.232564\pi$
$434$	18.5376	0.889832
$435$	0	0
$436$	−20.9141	−1.00161
$437$	1.41183	0.0675369
$438$	0	0
$439$	−13.0655	−0.623584	−0.311792	−	0.950150i	$-0.600929\pi$
$439$	−13.0655	−0.623584	−0.311792	+	0.950150i	$0.600929\pi$
$440$	1.45218	0.0692300
$441$	0	0
$442$	−60.5121	−2.87827
$443$	−36.6631	−1.74192	−0.870959	−	0.491356i	$-0.836501\pi$
$443$	−36.6631	−1.74192	−0.870959	+	0.491356i	$0.836501\pi$
$444$	0	0
$445$	1.99875	0.0947497
$446$	8.24919	0.390611
$447$	0	0
$448$	−5.34533	−0.252543
$449$	27.1080	1.27930	0.639652	−	0.768665i	$-0.279079\pi$
$449$	27.1080	1.27930	0.639652	+	0.768665i	$0.279079\pi$
$450$	0	0
$451$	24.0048	1.13034
$452$	26.0092	1.22337
$453$	0	0
$454$	−49.1803	−2.30814
$455$	3.75543	0.176057
$456$	0	0
$457$	14.1819	0.663403	0.331702	−	0.943384i	$-0.392377\pi$
$457$	14.1819	0.663403	0.331702	+	0.943384i	$0.392377\pi$
$458$	37.5427	1.75426
$459$	0	0
$460$	0.607398	0.0283201
$461$	−19.4475	−0.905759	−0.452880	−	0.891572i	$-0.649603\pi$
$461$	−19.4475	−0.905759	−0.452880	+	0.891572i	$0.649603\pi$
$462$	0	0
$463$	−6.42291	−0.298498	−0.149249	−	0.988800i	$-0.547686\pi$
$463$	−6.42291	−0.298498	−0.149249	+	0.988800i	$0.547686\pi$
$464$	−28.2008	−1.30919
$465$	0	0
$466$	−10.5031	−0.486544
$467$	−26.9168	−1.24556	−0.622781	−	0.782396i	$-0.713997\pi$
$467$	−26.9168	−1.24556	−0.622781	+	0.782396i	$0.713997\pi$
$468$	0	0
$469$	12.1648	0.561719
$470$	11.0431	0.509381
$471$	0	0
$472$	0.814183	0.0374758
$473$	−2.58524	−0.118869
$474$	0	0
$475$	13.1074	0.601410
$476$	10.6739	0.489238
$477$	0	0
$478$	−18.3102	−0.837487
$479$	26.9980	1.23357	0.616785	−	0.787132i	$-0.288435\pi$
$479$	26.9980	1.23357	0.616785	+	0.787132i	$0.288435\pi$
$480$	0	0
$481$	0.0125906	0.000574080 0
$482$	−44.5719	−2.03019
$483$	0	0
$484$	−1.03284	−0.0469474
$485$	−6.88602	−0.312678
$486$	0	0
$487$	2.34982	0.106481	0.0532404	−	0.998582i	$-0.483045\pi$
$487$	2.34982	0.106481	0.0532404	+	0.998582i	$0.483045\pi$
$488$	−6.54529	−0.296291
$489$	0	0
$490$	−1.44701	−0.0653693
$491$	36.6159	1.65245	0.826227	−	0.563338i	$-0.190483\pi$
$491$	36.6159	1.65245	0.826227	+	0.563338i	$0.190483\pi$
$492$	0	0
$493$	39.3873	1.77392
$494$	28.3098	1.27372
$495$	0	0
$496$	−43.6832	−1.96143
$497$	9.50724	0.426458
$498$	0	0
$499$	−25.6041	−1.14620	−0.573099	−	0.819486i	$-0.694259\pi$
$499$	−25.6041	−1.14620	−0.573099	+	0.819486i	$0.694259\pi$
$500$	12.0003	0.536672
$501$	0	0
$502$	−1.19938	−0.0535308
$503$	−34.8427	−1.55356	−0.776780	−	0.629772i	$-0.783148\pi$
$503$	−34.8427	−1.55356	−0.776780	+	0.629772i	$0.783148\pi$
$504$	0	0
$505$	5.47320	0.243554
$506$	−2.95533	−0.131380
$507$	0	0
$508$	1.68863	0.0749209
$509$	37.6676	1.66959	0.834794	−	0.550563i	$-0.185587\pi$
$509$	37.6676	1.66959	0.834794	+	0.550563i	$0.185587\pi$
$510$	0	0
$511$	−2.92872	−0.129559
$512$	28.5129	1.26011
$513$	0	0
$514$	−5.52795	−0.243827
$515$	13.4699	0.593557
$516$	0	0
$517$	−24.5976	−1.08180
$518$	−0.00485128	−0.000213153 0
$519$	0	0
$520$	−2.24579	−0.0984842
$521$	6.44530	0.282374	0.141187	−	0.989983i	$-0.454908\pi$
$521$	6.44530	0.282374	0.141187	+	0.989983i	$0.454908\pi$
$522$	0	0
$523$	30.8509	1.34902	0.674508	−	0.738268i	$-0.264356\pi$
$523$	30.8509	1.34902	0.674508	+	0.738268i	$0.264356\pi$
$524$	−20.3833	−0.890447
$525$	0	0
$526$	−13.9635	−0.608838
$527$	61.0111	2.65769
$528$	0	0
$529$	−22.7721	−0.990090
$530$	1.49625	0.0649930
$531$	0	0
$532$	−4.99364	−0.216502
$533$	−37.1232	−1.60798
$534$	0	0
$535$	1.07354	0.0464130
$536$	−7.27468	−0.314218
$537$	0	0
$538$	25.4066	1.09536
$539$	3.22310	0.138829
$540$	0	0
$541$	21.0659	0.905692	0.452846	−	0.891589i	$-0.350409\pi$
$541$	21.0659	0.905692	0.452846	+	0.891589i	$0.350409\pi$
$542$	−59.6284	−2.56126
$543$	0	0
$544$	−47.3833	−2.03154
$545$	9.33135	0.399711
$546$	0	0
$547$	−3.29972	−0.141086	−0.0705429	−	0.997509i	$-0.522473\pi$
$547$	−3.29972	−0.141086	−0.0705429	+	0.997509i	$0.522473\pi$
$548$	−1.87558	−0.0801209
$549$	0	0
$550$	−27.4372	−1.16993
$551$	−18.4268	−0.785009
$552$	0	0
$553$	14.7112	0.625586
$554$	3.33017	0.141485
$555$	0	0
$556$	−32.0562	−1.35949
$557$	−12.5616	−0.532251	−0.266125	−	0.963938i	$-0.585744\pi$
$557$	−12.5616	−0.532251	−0.266125	+	0.963938i	$0.585744\pi$
$558$	0	0
$559$	3.99805	0.169100
$560$	3.40983	0.144092
$561$	0	0
$562$	53.6011	2.26103
$563$	−0.665089	−0.0280302	−0.0140151	−	0.999902i	$-0.504461\pi$
$563$	−0.665089	−0.0280302	−0.0140151	+	0.999902i	$0.504461\pi$
$564$	0	0
$565$	−11.6046	−0.488211
$566$	−18.4261	−0.774506
$567$	0	0
$568$	−5.68542	−0.238555
$569$	28.0959	1.17784	0.588921	−	0.808190i	$-0.299553\pi$
$569$	28.0959	1.17784	0.588921	+	0.808190i	$0.299553\pi$
$570$	0	0
$571$	1.61819	0.0677191	0.0338596	−	0.999427i	$-0.489220\pi$
$571$	1.61819	0.0677191	0.0338596	+	0.999427i	$0.489220\pi$
$572$	−27.1287	−1.13431
$573$	0	0
$574$	14.3040	0.597037
$575$	2.11609	0.0882470
$576$	0	0
$577$	−40.8337	−1.69993	−0.849964	−	0.526840i	$-0.823377\pi$
$577$	−40.8337	−1.69993	−0.849964	+	0.526840i	$0.823377\pi$
$578$	44.0880	1.83382
$579$	0	0
$580$	−7.92760	−0.329176
$581$	12.3257	0.511356
$582$	0	0
$583$	−3.33278	−0.138029
$584$	1.75140	0.0724736
$585$	0	0
$586$	14.3731	0.593746
$587$	−21.6827	−0.894943	−0.447471	−	0.894298i	$-0.647675\pi$
$587$	−21.6827	−0.894943	−0.447471	+	0.894298i	$0.647675\pi$
$588$	0	0
$589$	−28.5432	−1.17610
$590$	1.97009	0.0811072
$591$	0	0
$592$	0.0114319	0.000469848 0
$593$	−36.0630	−1.48093	−0.740464	−	0.672096i	$-0.765394\pi$
$593$	−36.0630	−1.48093	−0.740464	+	0.672096i	$0.765394\pi$
$594$	0	0
$595$	−4.76242	−0.195240
$596$	10.3778	0.425091
$597$	0	0
$598$	4.57038	0.186897
$599$	−14.9942	−0.612648	−0.306324	−	0.951927i	$-0.599099\pi$
$599$	−14.9942	−0.612648	−0.306324	+	0.951927i	$0.599099\pi$
$600$	0	0
$601$	39.6897	1.61898	0.809489	−	0.587136i	$-0.199744\pi$
$601$	39.6897	1.61898	0.809489	+	0.587136i	$0.199744\pi$
$602$	−1.54049	−0.0627859
$603$	0	0
$604$	30.2305	1.23006
$605$	0.460828	0.0187353
$606$	0	0
$607$	−23.0549	−0.935768	−0.467884	−	0.883790i	$-0.654984\pi$
$607$	−23.0549	−0.935768	−0.467884	+	0.883790i	$0.654984\pi$
$608$	22.1676	0.899016
$609$	0	0
$610$	−15.8377	−0.641250
$611$	38.0400	1.53894
$612$	0	0
$613$	−20.5887	−0.831569	−0.415785	−	0.909463i	$-0.636493\pi$
$613$	−20.5887	−0.831569	−0.415785	+	0.909463i	$0.636493\pi$
$614$	−4.57927	−0.184804
$615$	0	0
$616$	−1.92744	−0.0776589
$617$	34.0369	1.37028	0.685138	−	0.728414i	$-0.259742\pi$
$617$	34.0369	1.37028	0.685138	+	0.728414i	$0.259742\pi$
$618$	0	0
$619$	14.7367	0.592320	0.296160	−	0.955138i	$-0.404294\pi$
$619$	14.7367	0.592320	0.296160	+	0.955138i	$0.404294\pi$
$620$	−12.2799	−0.493172
$621$	0	0
$622$	23.6712	0.949127
$623$	−2.65289	−0.106286
$624$	0	0
$625$	16.8075	0.672301
$626$	−33.3363	−1.33239
$627$	0	0
$628$	8.80028	0.351170
$629$	−0.0159666	−0.000636630 0
$630$	0	0
$631$	7.75176	0.308593	0.154296	−	0.988025i	$-0.450689\pi$
$631$	7.75176	0.308593	0.154296	+	0.988025i	$0.450689\pi$
$632$	−8.79747	−0.349945
$633$	0	0
$634$	16.3942	0.651097
$635$	−0.753423	−0.0298987
$636$	0	0
$637$	−4.98449	−0.197493
$638$	38.5721	1.52709
$639$	0	0
$640$	−3.56075	−0.140751
$641$	2.00673	0.0792612	0.0396306	−	0.999214i	$-0.487382\pi$
$641$	2.00673	0.0792612	0.0396306	+	0.999214i	$0.487382\pi$
$642$	0	0
$643$	−27.9572	−1.10252	−0.551262	−	0.834332i	$-0.685854\pi$
$643$	−27.9572	−1.10252	−0.551262	+	0.834332i	$0.685854\pi$
$644$	−0.806184	−0.0317681
$645$	0	0
$646$	−35.9008	−1.41250
$647$	−25.1931	−0.990444	−0.495222	−	0.868766i	$-0.664913\pi$
$647$	−25.1931	−0.990444	−0.495222	+	0.868766i	$0.664913\pi$
$648$	0	0
$649$	−4.38821	−0.172252
$650$	42.4314	1.66430
$651$	0	0
$652$	29.6124	1.15971
$653$	−40.1842	−1.57253	−0.786264	−	0.617891i	$-0.787987\pi$
$653$	−40.1842	−1.57253	−0.786264	+	0.617891i	$0.787987\pi$
$654$	0	0
$655$	9.09449	0.355351
$656$	−33.7069	−1.31603
$657$	0	0
$658$	−14.6573	−0.571399
$659$	−6.34563	−0.247191	−0.123595	−	0.992333i	$-0.539442\pi$
$659$	−6.34563	−0.247191	−0.123595	+	0.992333i	$0.539442\pi$
$660$	0	0
$661$	35.8318	1.39370	0.696849	−	0.717218i	$-0.254585\pi$
$661$	35.8318	1.39370	0.696849	+	0.717218i	$0.254585\pi$
$662$	−6.40611	−0.248981
$663$	0	0
$664$	−7.37089	−0.286046
$665$	2.22803	0.0863995
$666$	0	0
$667$	−2.97486	−0.115187
$668$	35.0155	1.35479
$669$	0	0
$670$	−17.6026	−0.680048
$671$	35.2772	1.36186
$672$	0	0
$673$	−17.9770	−0.692961	−0.346481	−	0.938057i	$-0.612623\pi$
$673$	−17.9770	−0.692961	−0.346481	+	0.938057i	$0.612623\pi$
$674$	44.9826	1.73266
$675$	0	0
$676$	20.0021	0.769311
$677$	7.99135	0.307133	0.153566	−	0.988138i	$-0.450924\pi$
$677$	7.99135	0.307133	0.153566	+	0.988138i	$0.450924\pi$
$678$	0	0
$679$	9.13964	0.350747
$680$	2.84797	0.109215
$681$	0	0
$682$	59.7484	2.28789
$683$	−17.7118	−0.677723	−0.338862	−	0.940836i	$-0.610042\pi$
$683$	−17.7118	−0.677723	−0.338862	+	0.940836i	$0.610042\pi$
$684$	0	0
$685$	0.836836	0.0319739
$686$	1.92058	0.0733281
$687$	0	0
$688$	3.63013	0.138397
$689$	5.15411	0.196356
$690$	0	0
$691$	−37.6270	−1.43140	−0.715700	−	0.698408i	$-0.753892\pi$
$691$	−37.6270	−1.43140	−0.715700	+	0.698408i	$0.753892\pi$
$692$	12.4260	0.472366
$693$	0	0
$694$	42.9771	1.63139
$695$	14.3027	0.542531
$696$	0	0
$697$	47.0774	1.78319
$698$	−2.81638	−0.106601
$699$	0	0
$700$	−7.48461	−0.282892
$701$	−15.0813	−0.569612	−0.284806	−	0.958585i	$-0.591929\pi$
$701$	−15.0813	−0.569612	−0.284806	+	0.958585i	$0.591929\pi$
$702$	0	0
$703$	0.00746977	0.000281728 0
$704$	−17.2285	−0.649324
$705$	0	0
$706$	−19.3935	−0.729883
$707$	−7.26444	−0.273207
$708$	0	0
$709$	6.88695	0.258645	0.129323	−	0.991603i	$-0.458720\pi$
$709$	6.88695	0.258645	0.129323	+	0.991603i	$0.458720\pi$
$710$	−13.7571	−0.516294
$711$	0	0
$712$	1.58645	0.0594548
$713$	−4.60808	−0.172574
$714$	0	0
$715$	12.1041	0.452668
$716$	14.5562	0.543989
$717$	0	0
$718$	66.2969	2.47418
$719$	12.7380	0.475047	0.237523	−	0.971382i	$-0.423664\pi$
$719$	12.7380	0.475047	0.237523	+	0.971382i	$0.423664\pi$
$720$	0	0
$721$	−17.8783	−0.665823
$722$	−19.6953	−0.732984
$723$	0	0
$724$	28.6299	1.06402
$725$	−27.6186	−1.02573
$726$	0	0
$727$	−5.23009	−0.193973	−0.0969867	−	0.995286i	$-0.530920\pi$
$727$	−5.23009	−0.193973	−0.0969867	+	0.995286i	$0.530920\pi$
$728$	2.98077	0.110475
$729$	0	0
$730$	4.23789	0.156851
$731$	−5.07009	−0.187524
$732$	0	0
$733$	−36.6517	−1.35376	−0.676880	−	0.736093i	$-0.736669\pi$
$733$	−36.6517	−1.35376	−0.676880	+	0.736093i	$0.736669\pi$
$734$	35.8419	1.32295
$735$	0	0
$736$	3.57879	0.131916
$737$	39.2084	1.44426
$738$	0	0
$739$	−20.1471	−0.741125	−0.370562	−	0.928808i	$-0.620835\pi$
$739$	−20.1471	−0.741125	−0.370562	+	0.928808i	$0.620835\pi$
$740$	0.00321364	0.000118136 0
$741$	0	0
$742$	−1.98594	−0.0729060
$743$	25.6183	0.939846	0.469923	−	0.882707i	$-0.344282\pi$
$743$	25.6183	0.939846	0.469923	+	0.882707i	$0.344282\pi$
$744$	0	0
$745$	−4.63030	−0.169641
$746$	−52.3711	−1.91744
$747$	0	0
$748$	34.4030	1.25790
$749$	−1.42488	−0.0520639
$750$	0	0
$751$	−44.2989	−1.61649	−0.808246	−	0.588846i	$-0.799583\pi$
$751$	−44.2989	−1.61649	−0.808246	+	0.588846i	$0.799583\pi$
$752$	34.5394	1.25952
$753$	0	0
$754$	−59.6515	−2.17238
$755$	−13.4881	−0.490881
$756$	0	0
$757$	−13.8335	−0.502788	−0.251394	−	0.967885i	$-0.580889\pi$
$757$	−13.8335	−0.502788	−0.251394	+	0.967885i	$0.580889\pi$
$758$	−64.2154	−2.33241
$759$	0	0
$760$	−1.33239	−0.0483307
$761$	−50.8106	−1.84188	−0.920941	−	0.389703i	$-0.872578\pi$
$761$	−50.8106	−1.84188	−0.920941	+	0.389703i	$0.872578\pi$
$762$	0	0
$763$	−12.3853	−0.448377
$764$	26.4169	0.955731
$765$	0	0
$766$	−32.2619	−1.16567
$767$	6.78632	0.245040
$768$	0	0
$769$	39.8702	1.43776	0.718878	−	0.695136i	$-0.244656\pi$
$769$	39.8702	1.43776	0.718878	+	0.695136i	$0.244656\pi$
$770$	−4.66385	−0.168074
$771$	0	0
$772$	−7.37311	−0.265364
$773$	−6.87681	−0.247342	−0.123671	−	0.992323i	$-0.539467\pi$
$773$	−6.87681	−0.247342	−0.123671	+	0.992323i	$0.539467\pi$
$774$	0	0
$775$	−42.7814	−1.53675
$776$	−5.46560	−0.196204
$777$	0	0
$778$	−28.8098	−1.03288
$779$	−22.0246	−0.789112
$780$	0	0
$781$	30.6427	1.09648
$782$	−5.79589	−0.207261
$783$	0	0
$784$	−4.52579	−0.161635
$785$	−3.92646	−0.140141
$786$	0	0
$787$	5.18048	0.184664	0.0923321	−	0.995728i	$-0.470568\pi$
$787$	5.18048	0.184664	0.0923321	+	0.995728i	$0.470568\pi$
$788$	−15.8990	−0.566377
$789$	0	0
$790$	−21.2873	−0.757369
$791$	15.4026	0.547652
$792$	0	0
$793$	−54.5559	−1.93734
$794$	−5.49146	−0.194885
$795$	0	0
$796$	−35.1754	−1.24676
$797$	33.9837	1.20376	0.601882	−	0.798585i	$-0.294418\pi$
$797$	33.9837	1.20376	0.601882	+	0.798585i	$0.294418\pi$
$798$	0	0
$799$	−48.2402	−1.70661
$800$	33.2255	1.17470
$801$	0	0
$802$	−1.47012	−0.0519118
$803$	−9.43955	−0.333115
$804$	0	0
$805$	0.359698	0.0126777
$806$	−92.4004	−3.25467
$807$	0	0
$808$	4.34421	0.152829
$809$	−15.8374	−0.556815	−0.278407	−	0.960463i	$-0.589806\pi$
$809$	−15.8374	−0.556815	−0.278407	+	0.960463i	$0.589806\pi$
$810$	0	0
$811$	−14.4073	−0.505909	−0.252954	−	0.967478i	$-0.581402\pi$
$811$	−14.4073	−0.505909	−0.252954	+	0.967478i	$0.581402\pi$
$812$	10.5221	0.369253
$813$	0	0
$814$	−0.0156362	−0.000548047 0
$815$	−13.2123	−0.462807
$816$	0	0
$817$	2.37198	0.0829849
$818$	−2.91744	−0.102006
$819$	0	0
$820$	−9.47541	−0.330896
$821$	−38.0778	−1.32892	−0.664462	−	0.747322i	$-0.731339\pi$
$821$	−38.0778	−1.32892	−0.664462	+	0.747322i	$0.731339\pi$
$822$	0	0
$823$	−14.6464	−0.510541	−0.255271	−	0.966870i	$-0.582165\pi$
$823$	−14.6464	−0.510541	−0.255271	+	0.966870i	$0.582165\pi$
$824$	10.6914	0.372453
$825$	0	0
$826$	−2.61485	−0.0909822
$827$	39.9464	1.38907	0.694535	−	0.719459i	$-0.255610\pi$
$827$	39.9464	1.38907	0.694535	+	0.719459i	$0.255610\pi$
$828$	0	0
$829$	37.0700	1.28750	0.643748	−	0.765238i	$-0.277379\pi$
$829$	37.0700	1.28750	0.643748	+	0.765238i	$0.277379\pi$
$830$	−17.8354	−0.619076
$831$	0	0
$832$	26.6438	0.923706
$833$	6.32104	0.219011
$834$	0	0
$835$	−15.6230	−0.540657
$836$	−16.0950	−0.556657
$837$	0	0
$838$	−5.74258	−0.198374
$839$	−16.4152	−0.566717	−0.283359	−	0.959014i	$-0.591449\pi$
$839$	−16.4152	−0.566717	−0.283359	+	0.959014i	$0.591449\pi$
$840$	0	0
$841$	9.82716	0.338867
$842$	−41.3780	−1.42598
$843$	0	0
$844$	40.9182	1.40846
$845$	−8.92441	−0.307009
$846$	0	0
$847$	−0.611645	−0.0210164
$848$	4.67980	0.160705
$849$	0	0
$850$	−53.8091	−1.84564
$851$	0.00120593	4.13389e−5 0
$852$	0	0
$853$	11.6795	0.399900	0.199950	−	0.979806i	$-0.435922\pi$
$853$	11.6795	0.399900	0.199950	+	0.979806i	$0.435922\pi$
$854$	21.0210	0.719324
$855$	0	0
$856$	0.852091	0.0291239
$857$	−51.6216	−1.76336	−0.881681	−	0.471846i	$-0.843588\pi$
$857$	−51.6216	−1.76336	−0.881681	+	0.471846i	$0.843588\pi$
$858$	0	0
$859$	32.6184	1.11293	0.556463	−	0.830873i	$-0.312158\pi$
$859$	32.6184	1.11293	0.556463	+	0.830873i	$0.312158\pi$
$860$	1.02047	0.0347978
$861$	0	0
$862$	34.8506	1.18701
$863$	−31.9589	−1.08789	−0.543946	−	0.839120i	$-0.683070\pi$
$863$	−31.9589	−1.08789	−0.543946	+	0.839120i	$0.683070\pi$
$864$	0	0
$865$	−5.54416	−0.188507
$866$	59.5281	2.02285
$867$	0	0
$868$	16.2988	0.553217
$869$	47.4158	1.60847
$870$	0	0
$871$	−60.6354	−2.05455
$872$	7.40651	0.250816
$873$	0	0
$874$	2.71153	0.0917189
$875$	7.10655	0.240245
$876$	0	0
$877$	24.0574	0.812361	0.406180	−	0.913793i	$-0.366861\pi$
$877$	24.0574	0.812361	0.406180	+	0.913793i	$0.366861\pi$
$878$	−25.0934	−0.846862
$879$	0	0
$880$	10.9902	0.370480
$881$	5.78624	0.194943	0.0974717	−	0.995238i	$-0.468924\pi$
$881$	5.78624	0.194943	0.0974717	+	0.995238i	$0.468924\pi$
$882$	0	0
$883$	−14.0643	−0.473301	−0.236651	−	0.971595i	$-0.576050\pi$
$883$	−14.0643	−0.473301	−0.236651	+	0.971595i	$0.576050\pi$
$884$	−53.2040	−1.78944
$885$	0	0
$886$	−70.4145	−2.36562
$887$	48.1408	1.61641	0.808204	−	0.588902i	$-0.200440\pi$
$887$	48.1408	1.61641	0.808204	+	0.588902i	$0.200440\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	3.83875	0.128675
$891$	0	0
$892$	7.25293	0.242846
$893$	22.5685	0.755226
$894$	0	0
$895$	−6.49458	−0.217090
$896$	4.72610	0.157888
$897$	0	0
$898$	52.0630	1.73737
$899$	60.1434	2.00590
$900$	0	0
$901$	−6.53614	−0.217750
$902$	46.1031	1.53507
$903$	0	0
$904$	−9.21088	−0.306349
$905$	−12.7739	−0.424619
$906$	0	0
$907$	25.5385	0.847990	0.423995	−	0.905664i	$-0.360627\pi$
$907$	25.5385	0.847990	0.423995	+	0.905664i	$0.360627\pi$
$908$	−43.2407	−1.43499
$909$	0	0
$910$	7.21261	0.239096
$911$	−3.33562	−0.110514	−0.0552571	−	0.998472i	$-0.517598\pi$
$911$	−3.33562	−0.110514	−0.0552571	+	0.998472i	$0.517598\pi$
$912$	0	0
$913$	39.7269	1.31477
$914$	27.2376	0.900938
$915$	0	0
$916$	33.0087	1.09064
$917$	−12.0709	−0.398616
$918$	0	0
$919$	9.18196	0.302885	0.151442	−	0.988466i	$-0.451608\pi$
$919$	9.18196	0.302885	0.151442	+	0.988466i	$0.451608\pi$
$920$	−0.215103	−0.00709174
$921$	0	0
$922$	−37.3504	−1.23007
$923$	−47.3887	−1.55982
$924$	0	0
$925$	0.0111959	0.000368119 0
$926$	−12.3357	−0.405377
$927$	0	0
$928$	−46.7094	−1.53331
$929$	8.62340	0.282924	0.141462	−	0.989944i	$-0.454820\pi$
$929$	8.62340	0.282924	0.141462	+	0.989944i	$0.454820\pi$
$930$	0	0
$931$	−2.95721	−0.0969188
$932$	−9.23459	−0.302489
$933$	0	0
$934$	−51.6960	−1.69154
$935$	−15.3497	−0.501990
$936$	0	0
$937$	29.8031	0.973624	0.486812	−	0.873507i	$-0.338160\pi$
$937$	29.8031	0.973624	0.486812	+	0.873507i	$0.338160\pi$
$938$	23.3635	0.762846
$939$	0	0
$940$	9.70943	0.316687
$941$	8.28031	0.269930	0.134965	−	0.990850i	$-0.456908\pi$
$941$	8.28031	0.269930	0.134965	+	0.990850i	$0.456908\pi$
$942$	0	0
$943$	−3.55569	−0.115789
$944$	6.16180	0.200550
$945$	0	0
$946$	−4.96516	−0.161431
$947$	−33.2181	−1.07944	−0.539722	−	0.841844i	$-0.681470\pi$
$947$	−33.2181	−1.07944	−0.539722	+	0.841844i	$0.681470\pi$
$948$	0	0
$949$	14.5982	0.473877
$950$	25.1739	0.816748
$951$	0	0
$952$	−3.78004	−0.122512
$953$	45.0975	1.46085	0.730425	−	0.682993i	$-0.239322\pi$
$953$	45.0975	1.46085	0.730425	+	0.682993i	$0.239322\pi$
$954$	0	0
$955$	−11.7866	−0.381404
$956$	−16.0988	−0.520673
$957$	0	0
$958$	51.8518	1.67526
$959$	−1.11071	−0.0358668
$960$	0	0
$961$	62.1625	2.00524
$962$	0.0241812	0.000779633 0
$963$	0	0
$964$	−39.1889	−1.26219
$965$	3.28969	0.105899
$966$	0	0
$967$	−51.2168	−1.64702	−0.823511	−	0.567300i	$-0.807988\pi$
$967$	−51.2168	−1.64702	−0.823511	+	0.567300i	$0.807988\pi$
$968$	0.365770	0.0117563
$969$	0	0
$970$	−13.2252	−0.424634
$971$	−12.2979	−0.394658	−0.197329	−	0.980337i	$-0.563227\pi$
$971$	−12.2979	−0.394658	−0.197329	+	0.980337i	$0.563227\pi$
$972$	0	0
$973$	−18.9836	−0.608585
$974$	4.51303	0.144607
$975$	0	0
$976$	−49.5353	−1.58559
$977$	−9.71148	−0.310698	−0.155349	−	0.987860i	$-0.549650\pi$
$977$	−9.71148	−0.310698	−0.155349	+	0.987860i	$0.549650\pi$
$978$	0	0
$979$	−8.55051	−0.273275
$980$	−1.27225	−0.0406407
$981$	0	0
$982$	70.3238	2.24412
$983$	−12.7440	−0.406471	−0.203236	−	0.979130i	$-0.565146\pi$
$983$	−12.7440	−0.406471	−0.203236	+	0.979130i	$0.565146\pi$
$984$	0	0
$985$	7.09370	0.226024
$986$	75.6465	2.40908
$987$	0	0
$988$	24.8908	0.791881
$989$	0.382937	0.0121767
$990$	0	0
$991$	1.52505	0.0484450	0.0242225	−	0.999707i	$-0.492289\pi$
$991$	1.52505	0.0484450	0.0242225	+	0.999707i	$0.492289\pi$
$992$	−72.3531	−2.29721
$993$	0	0
$994$	18.2594	0.579153
$995$	15.6944	0.497545
$996$	0	0
$997$	−52.5315	−1.66369	−0.831844	−	0.555009i	$-0.812715\pi$
$997$	−52.5315	−1.66369	−0.831844	+	0.555009i	$0.812715\pi$
$998$	−49.1748	−1.55660
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.n.1.9		12
3.2	odd	2		889.2.a.a.1.4	✓	12
21.20	even	2		6223.2.a.i.1.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.a.1.4	✓	12	3.2	odd	2
6223.2.a.i.1.4		12	21.20	even	2
8001.2.a.n.1.9		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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+1.92058 q^{2} +1.68863 q^{4} -0.753423 q^{5} +1.00000 q^{7} -0.598010 q^{8} +O(q^{10})\)	\(q+1.92058 q^{2} +1.68863 q^{4} -0.753423 q^{5} +1.00000 q^{7} -0.598010 q^{8} -1.44701 q^{10} +3.22310 q^{11} -4.98449 q^{13} +1.92058 q^{14} -4.52579 q^{16} +6.32104 q^{17} -2.95721 q^{19} -1.27225 q^{20} +6.19022 q^{22} -0.477419 q^{23} -4.43235 q^{25} -9.57312 q^{26} +1.68863 q^{28} +6.23114 q^{29} +9.65207 q^{31} -7.49612 q^{32} +12.1401 q^{34} -0.753423 q^{35} -0.00252595 q^{37} -5.67957 q^{38} +0.450555 q^{40} +7.44774 q^{41} -0.802098 q^{43} +5.44262 q^{44} -0.916921 q^{46} -7.63168 q^{47} +1.00000 q^{49} -8.51269 q^{50} -8.41696 q^{52} -1.03403 q^{53} -2.42836 q^{55} -0.598010 q^{56} +11.9674 q^{58} -1.36149 q^{59} +10.9451 q^{61} +18.5376 q^{62} -5.34533 q^{64} +3.75543 q^{65} +12.1648 q^{67} +10.6739 q^{68} -1.44701 q^{70} +9.50724 q^{71} -2.92872 q^{73} -0.00485128 q^{74} -4.99364 q^{76} +3.22310 q^{77} +14.7112 q^{79} +3.40983 q^{80} +14.3040 q^{82} +12.3257 q^{83} -4.76242 q^{85} -1.54049 q^{86} -1.92744 q^{88} -2.65289 q^{89} -4.98449 q^{91} -0.806184 q^{92} -14.6573 q^{94} +2.22803 q^{95} +9.13964 q^{97} +1.92058 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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