LMFDB - Embedded newform 8001.2.a.w.1.20

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:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} + \cdots + 31 $ x^20 - 8x^19 + 152x^17 - 274x^16 - 1061x^15 + 3125x^14 + 2977x^13 - 15474x^12 - 56x^11 + 39579x^10 - 17664x^9 - 52271x^8 + 35701x^7 + 32493x^6 - 25504x^5 - 8607x^4 + 6812x^3 + 609x^2 - 425x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Root			$-2.48579$ 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)

$f(q)$	$=$	$q+2.48579 q^{2} +4.17918 q^{4} -3.35246 q^{5} +1.00000 q^{7} +5.41698 q^{8} +O(q^{10})$	\(q+2.48579 q^{2} +4.17918 q^{4} -3.35246 q^{5} +1.00000 q^{7} +5.41698 q^{8} -8.33354 q^{10} -1.56393 q^{11} +0.560792 q^{13} +2.48579 q^{14} +5.10716 q^{16} +1.90633 q^{17} +3.15361 q^{19} -14.0105 q^{20} -3.88760 q^{22} -7.59584 q^{23} +6.23902 q^{25} +1.39401 q^{26} +4.17918 q^{28} -7.34983 q^{29} -3.72875 q^{31} +1.86138 q^{32} +4.73874 q^{34} -3.35246 q^{35} -4.98201 q^{37} +7.83922 q^{38} -18.1602 q^{40} -6.64574 q^{41} -1.33203 q^{43} -6.53592 q^{44} -18.8817 q^{46} +2.63698 q^{47} +1.00000 q^{49} +15.5089 q^{50} +2.34365 q^{52} +2.87115 q^{53} +5.24301 q^{55} +5.41698 q^{56} -18.2702 q^{58} +11.5527 q^{59} +0.252927 q^{61} -9.26890 q^{62} -5.58731 q^{64} -1.88004 q^{65} -1.89376 q^{67} +7.96688 q^{68} -8.33354 q^{70} -7.84857 q^{71} +10.3325 q^{73} -12.3843 q^{74} +13.1795 q^{76} -1.56393 q^{77} -1.28541 q^{79} -17.1216 q^{80} -16.5199 q^{82} -9.61160 q^{83} -6.39090 q^{85} -3.31116 q^{86} -8.47177 q^{88} +1.02178 q^{89} +0.560792 q^{91} -31.7443 q^{92} +6.55498 q^{94} -10.5724 q^{95} -0.963940 q^{97} +2.48579 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * 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 $

$2$	2.48579	1.75772	0.878861	−	0.477078i	$-0.158304\pi$
$2$	2.48579	1.75772	0.878861	+	0.477078i	$0.158304\pi$
$3$	0	0
$3$	0	0
$4$	4.17918	2.08959
$5$	−3.35246	−1.49927	−0.749634	−	0.661853i	$-0.769770\pi$
$5$	−3.35246	−1.49927	−0.749634	+	0.661853i	$0.769770\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	5.41698	1.91519
$9$	0	0
$10$	−8.33354	−2.63530
$11$	−1.56393	−0.471542	−0.235771	−	0.971809i	$-0.575761\pi$
$11$	−1.56393	−0.471542	−0.235771	+	0.971809i	$0.575761\pi$
$12$	0	0
$13$	0.560792	0.155536	0.0777679	−	0.996971i	$-0.475221\pi$
$13$	0.560792	0.155536	0.0777679	+	0.996971i	$0.475221\pi$
$14$	2.48579	0.664357
$15$	0	0
$16$	5.10716	1.27679
$17$	1.90633	0.462352	0.231176	−	0.972912i	$-0.425743\pi$
$17$	1.90633	0.462352	0.231176	+	0.972912i	$0.425743\pi$
$18$	0	0
$19$	3.15361	0.723487	0.361743	−	0.932278i	$-0.382182\pi$
$19$	3.15361	0.723487	0.361743	+	0.932278i	$0.382182\pi$
$20$	−14.0105	−3.13285
$21$	0	0
$22$	−3.88760	−0.828839
$23$	−7.59584	−1.58384	−0.791921	−	0.610624i	$-0.790919\pi$
$23$	−7.59584	−1.58384	−0.791921	+	0.610624i	$0.790919\pi$
$24$	0	0
$25$	6.23902	1.24780
$26$	1.39401	0.273389
$27$	0	0
$28$	4.17918	0.789790
$29$	−7.34983	−1.36483	−0.682415	−	0.730965i	$-0.739070\pi$
$29$	−7.34983	−1.36483	−0.682415	+	0.730965i	$0.739070\pi$
$30$	0	0
$31$	−3.72875	−0.669703	−0.334851	−	0.942271i	$-0.608686\pi$
$31$	−3.72875	−0.669703	−0.334851	+	0.942271i	$0.608686\pi$
$32$	1.86138	0.329048
$33$	0	0
$34$	4.73874	0.812687
$35$	−3.35246	−0.566670
$36$	0	0
$37$	−4.98201	−0.819038	−0.409519	−	0.912302i	$-0.634303\pi$
$37$	−4.98201	−0.819038	−0.409519	+	0.912302i	$0.634303\pi$
$38$	7.83922	1.27169
$39$	0	0
$40$	−18.1602	−2.87139
$41$	−6.64574	−1.03789	−0.518945	−	0.854808i	$-0.673675\pi$
$41$	−6.64574	−1.03789	−0.518945	+	0.854808i	$0.673675\pi$
$42$	0	0
$43$	−1.33203	−0.203133	−0.101567	−	0.994829i	$-0.532386\pi$
$43$	−1.33203	−0.203133	−0.101567	+	0.994829i	$0.532386\pi$
$44$	−6.53592	−0.985328
$45$	0	0
$46$	−18.8817	−2.78395
$47$	2.63698	0.384642	0.192321	−	0.981332i	$-0.438398\pi$
$47$	2.63698	0.384642	0.192321	+	0.981332i	$0.438398\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	15.5089	2.19329
$51$	0	0
$52$	2.34365	0.325006
$53$	2.87115	0.394383	0.197191	−	0.980365i	$-0.436818\pi$
$53$	2.87115	0.394383	0.197191	+	0.980365i	$0.436818\pi$
$54$	0	0
$55$	5.24301	0.706967
$56$	5.41698	0.723875
$57$	0	0
$58$	−18.2702	−2.39899
$59$	11.5527	1.50403	0.752017	−	0.659144i	$-0.229081\pi$
$59$	11.5527	1.50403	0.752017	+	0.659144i	$0.229081\pi$
$60$	0	0
$61$	0.252927	0.0323840	0.0161920	−	0.999869i	$-0.494846\pi$
$61$	0.252927	0.0323840	0.0161920	+	0.999869i	$0.494846\pi$
$62$	−9.26890	−1.17715
$63$	0	0
$64$	−5.58731	−0.698414
$65$	−1.88004	−0.233190
$66$	0	0
$67$	−1.89376	−0.231359	−0.115680	−	0.993287i	$-0.536905\pi$
$67$	−1.89376	−0.231359	−0.115680	+	0.993287i	$0.536905\pi$
$68$	7.96688	0.966126
$69$	0	0
$70$	−8.33354	−0.996048
$71$	−7.84857	−0.931454	−0.465727	−	0.884929i	$-0.654207\pi$
$71$	−7.84857	−0.931454	−0.465727	+	0.884929i	$0.654207\pi$
$72$	0	0
$73$	10.3325	1.20933	0.604664	−	0.796481i	$-0.293307\pi$
$73$	10.3325	1.20933	0.604664	+	0.796481i	$0.293307\pi$
$74$	−12.3843	−1.43964
$75$	0	0
$76$	13.1795	1.51179
$77$	−1.56393	−0.178226
$78$	0	0
$79$	−1.28541	−0.144620	−0.0723099	−	0.997382i	$-0.523037\pi$
$79$	−1.28541	−0.144620	−0.0723099	+	0.997382i	$0.523037\pi$
$80$	−17.1216	−1.91425
$81$	0	0
$82$	−16.5199	−1.82432
$83$	−9.61160	−1.05501	−0.527505	−	0.849552i	$-0.676872\pi$
$83$	−9.61160	−1.05501	−0.527505	+	0.849552i	$0.676872\pi$
$84$	0	0
$85$	−6.39090	−0.693190
$86$	−3.31116	−0.357052
$87$	0	0
$88$	−8.47177	−0.903093
$89$	1.02178	0.108309	0.0541543	−	0.998533i	$-0.482754\pi$
$89$	1.02178	0.108309	0.0541543	+	0.998533i	$0.482754\pi$
$90$	0	0
$91$	0.560792	0.0587870
$92$	−31.7443	−3.30958
$93$	0	0
$94$	6.55498	0.676094
$95$	−10.5724	−1.08470
$96$	0	0
$97$	−0.963940	−0.0978733	−0.0489367	−	0.998802i	$-0.515583\pi$
$97$	−0.963940	−0.0978733	−0.0489367	+	0.998802i	$0.515583\pi$
$98$	2.48579	0.251103
$99$	0	0
$100$	26.0740	2.60740
$101$	−11.4022	−1.13456	−0.567279	−	0.823526i	$-0.692004\pi$
$101$	−11.4022	−1.13456	−0.567279	+	0.823526i	$0.692004\pi$
$102$	0	0
$103$	−17.5045	−1.72477	−0.862384	−	0.506255i	$-0.831029\pi$
$103$	−17.5045	−1.72477	−0.862384	+	0.506255i	$0.831029\pi$
$104$	3.03780	0.297881
$105$	0	0
$106$	7.13709	0.693215
$107$	6.07352	0.587150	0.293575	−	0.955936i	$-0.405155\pi$
$107$	6.07352	0.587150	0.293575	+	0.955936i	$0.405155\pi$
$108$	0	0
$109$	−5.32495	−0.510038	−0.255019	−	0.966936i	$-0.582082\pi$
$109$	−5.32495	−0.510038	−0.255019	+	0.966936i	$0.582082\pi$
$110$	13.0330	1.24265
$111$	0	0
$112$	5.10716	0.482581
$113$	−16.4435	−1.54687	−0.773437	−	0.633874i	$-0.781464\pi$
$113$	−16.4435	−1.54687	−0.773437	+	0.633874i	$0.781464\pi$
$114$	0	0
$115$	25.4648	2.37460
$116$	−30.7162	−2.85193
$117$	0	0
$118$	28.7177	2.64367
$119$	1.90633	0.174753
$120$	0	0
$121$	−8.55413	−0.777648
$122$	0.628725	0.0569221
$123$	0	0
$124$	−15.5831	−1.39940
$125$	−4.15377	−0.371524
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−17.6117	−1.55667
$129$	0	0
$130$	−4.67338	−0.409883
$131$	−11.7524	−1.02681	−0.513404	−	0.858147i	$-0.671616\pi$
$131$	−11.7524	−1.02681	−0.513404	+	0.858147i	$0.671616\pi$
$132$	0	0
$133$	3.15361	0.273452
$134$	−4.70749	−0.406665
$135$	0	0
$136$	10.3265	0.885494
$137$	13.7085	1.17119	0.585597	−	0.810602i	$-0.300860\pi$
$137$	13.7085	1.17119	0.585597	+	0.810602i	$0.300860\pi$
$138$	0	0
$139$	18.5651	1.57467	0.787335	−	0.616525i	$-0.211460\pi$
$139$	18.5651	1.57467	0.787335	+	0.616525i	$0.211460\pi$
$140$	−14.0105	−1.18411
$141$	0	0
$142$	−19.5099	−1.63724
$143$	−0.877038	−0.0733416
$144$	0	0
$145$	24.6400	2.04624
$146$	25.6845	2.12566
$147$	0	0
$148$	−20.8207	−1.71145
$149$	14.7979	1.21229	0.606146	−	0.795354i	$-0.292715\pi$
$149$	14.7979	1.21229	0.606146	+	0.795354i	$0.292715\pi$
$150$	0	0
$151$	11.4934	0.935317	0.467658	−	0.883909i	$-0.345098\pi$
$151$	11.4934	0.935317	0.467658	+	0.883909i	$0.345098\pi$
$152$	17.0830	1.38562
$153$	0	0
$154$	−3.88760	−0.313272
$155$	12.5005	1.00406
$156$	0	0
$157$	−17.2200	−1.37431	−0.687153	−	0.726513i	$-0.741140\pi$
$157$	−17.2200	−1.37431	−0.687153	+	0.726513i	$0.741140\pi$
$158$	−3.19526	−0.254202
$159$	0	0
$160$	−6.24020	−0.493331
$161$	−7.59584	−0.598636
$162$	0	0
$163$	1.22641	0.0960599	0.0480299	−	0.998846i	$-0.484706\pi$
$163$	1.22641	0.0960599	0.0480299	+	0.998846i	$0.484706\pi$
$164$	−27.7737	−2.16876
$165$	0	0
$166$	−23.8925	−1.85441
$167$	8.30479	0.642644	0.321322	−	0.946970i	$-0.395873\pi$
$167$	8.30479	0.642644	0.321322	+	0.946970i	$0.395873\pi$
$168$	0	0
$169$	−12.6855	−0.975809
$170$	−15.8865	−1.21844
$171$	0	0
$172$	−5.56680	−0.424465
$173$	6.01867	0.457591	0.228796	−	0.973474i	$-0.426521\pi$
$173$	6.01867	0.457591	0.228796	+	0.973474i	$0.426521\pi$
$174$	0	0
$175$	6.23902	0.471625
$176$	−7.98722	−0.602059
$177$	0	0
$178$	2.53994	0.190376
$179$	−20.2061	−1.51028	−0.755138	−	0.655566i	$-0.772430\pi$
$179$	−20.2061	−1.51028	−0.755138	+	0.655566i	$0.772430\pi$
$180$	0	0
$181$	−12.7951	−0.951053	−0.475527	−	0.879701i	$-0.657742\pi$
$181$	−12.7951	−0.951053	−0.475527	+	0.879701i	$0.657742\pi$
$182$	1.39401	0.103331
$183$	0	0
$184$	−41.1465	−3.03336
$185$	16.7020	1.22796
$186$	0	0
$187$	−2.98136	−0.218018
$188$	11.0204	0.803744
$189$	0	0
$190$	−26.2807	−1.90660
$191$	−16.2318	−1.17449	−0.587247	−	0.809408i	$-0.699788\pi$
$191$	−16.2318	−1.17449	−0.587247	+	0.809408i	$0.699788\pi$
$192$	0	0
$193$	−24.2695	−1.74696	−0.873479	−	0.486862i	$-0.838141\pi$
$193$	−24.2695	−1.74696	−0.873479	+	0.486862i	$0.838141\pi$
$194$	−2.39616	−0.172034
$195$	0	0
$196$	4.17918	0.298513
$197$	−6.58820	−0.469390	−0.234695	−	0.972069i	$-0.575409\pi$
$197$	−6.58820	−0.469390	−0.234695	+	0.972069i	$0.575409\pi$
$198$	0	0
$199$	17.5920	1.24707	0.623533	−	0.781797i	$-0.285697\pi$
$199$	17.5920	1.24707	0.623533	+	0.781797i	$0.285697\pi$
$200$	33.7967	2.38978
$201$	0	0
$202$	−28.3434	−1.99424
$203$	−7.34983	−0.515857
$204$	0	0
$205$	22.2796	1.55607
$206$	−43.5125	−3.03166
$207$	0	0
$208$	2.86405	0.198586
$209$	−4.93201	−0.341154
$210$	0	0
$211$	−9.19907	−0.633290	−0.316645	−	0.948544i	$-0.602556\pi$
$211$	−9.19907	−0.633290	−0.316645	+	0.948544i	$0.602556\pi$
$212$	11.9990	0.824097
$213$	0	0
$214$	15.0975	1.03205
$215$	4.46560	0.304551
$216$	0	0
$217$	−3.72875	−0.253124
$218$	−13.2367	−0.896505
$219$	0	0
$220$	21.9115	1.47727
$221$	1.06905	0.0719123
$222$	0	0
$223$	−12.0425	−0.806424	−0.403212	−	0.915107i	$-0.632106\pi$
$223$	−12.0425	−0.806424	−0.403212	+	0.915107i	$0.632106\pi$
$224$	1.86138	0.124368
$225$	0	0
$226$	−40.8751	−2.71897
$227$	−2.27127	−0.150749	−0.0753747	−	0.997155i	$-0.524015\pi$
$227$	−2.27127	−0.150749	−0.0753747	+	0.997155i	$0.524015\pi$
$228$	0	0
$229$	11.6422	0.769336	0.384668	−	0.923055i	$-0.374316\pi$
$229$	11.6422	0.769336	0.384668	+	0.923055i	$0.374316\pi$
$230$	63.3002	4.17389
$231$	0	0
$232$	−39.8139	−2.61391
$233$	12.1608	0.796680	0.398340	−	0.917238i	$-0.369586\pi$
$233$	12.1608	0.796680	0.398340	+	0.917238i	$0.369586\pi$
$234$	0	0
$235$	−8.84037	−0.576682
$236$	48.2808	3.14281
$237$	0	0
$238$	4.73874	0.307167
$239$	−7.95463	−0.514542	−0.257271	−	0.966339i	$-0.582823\pi$
$239$	−7.95463	−0.514542	−0.257271	+	0.966339i	$0.582823\pi$
$240$	0	0
$241$	9.79797	0.631143	0.315571	−	0.948902i	$-0.397804\pi$
$241$	9.79797	0.631143	0.315571	+	0.948902i	$0.397804\pi$
$242$	−21.2638	−1.36689
$243$	0	0
$244$	1.05703	0.0676692
$245$	−3.35246	−0.214181
$246$	0	0
$247$	1.76852	0.112528
$248$	−20.1986	−1.28261
$249$	0	0
$250$	−10.3254	−0.653036
$251$	−15.7157	−0.991968	−0.495984	−	0.868332i	$-0.665193\pi$
$251$	−15.7157	−0.991968	−0.495984	+	0.868332i	$0.665193\pi$
$252$	0	0
$253$	11.8793	0.746847
$254$	−2.48579	−0.155973
$255$	0	0
$256$	−32.6044	−2.03777
$257$	20.4161	1.27352	0.636761	−	0.771061i	$-0.280274\pi$
$257$	20.4161	1.27352	0.636761	+	0.771061i	$0.280274\pi$
$258$	0	0
$259$	−4.98201	−0.309567
$260$	−7.85700	−0.487270
$261$	0	0
$262$	−29.2139	−1.80484
$263$	−11.4806	−0.707923	−0.353961	−	0.935260i	$-0.615166\pi$
$263$	−11.4806	−0.707923	−0.353961	+	0.935260i	$0.615166\pi$
$264$	0	0
$265$	−9.62543	−0.591285
$266$	7.83922	0.480653
$267$	0	0
$268$	−7.91434	−0.483445
$269$	−6.53180	−0.398251	−0.199125	−	0.979974i	$-0.563810\pi$
$269$	−6.53180	−0.398251	−0.199125	+	0.979974i	$0.563810\pi$
$270$	0	0
$271$	−16.6906	−1.01388	−0.506940	−	0.861982i	$-0.669223\pi$
$271$	−16.6906	−1.01388	−0.506940	+	0.861982i	$0.669223\pi$
$272$	9.73591	0.590326
$273$	0	0
$274$	34.0765	2.05863
$275$	−9.75737	−0.588391
$276$	0	0
$277$	−15.2783	−0.917985	−0.458992	−	0.888440i	$-0.651790\pi$
$277$	−15.2783	−0.917985	−0.458992	+	0.888440i	$0.651790\pi$
$278$	46.1490	2.76783
$279$	0	0
$280$	−18.1602	−1.08528
$281$	19.1608	1.14304	0.571518	−	0.820589i	$-0.306355\pi$
$281$	19.1608	1.14304	0.571518	+	0.820589i	$0.306355\pi$
$282$	0	0
$283$	−26.4492	−1.57224	−0.786120	−	0.618074i	$-0.787913\pi$
$283$	−26.4492	−1.57224	−0.786120	+	0.618074i	$0.787913\pi$
$284$	−32.8005	−1.94635
$285$	0	0
$286$	−2.18014	−0.128914
$287$	−6.64574	−0.392286
$288$	0	0
$289$	−13.3659	−0.786230
$290$	61.2501	3.59673
$291$	0	0
$292$	43.1814	2.52700
$293$	21.8930	1.27900	0.639502	−	0.768789i	$-0.279141\pi$
$293$	21.8930	1.27900	0.639502	+	0.768789i	$0.279141\pi$
$294$	0	0
$295$	−38.7300	−2.25495
$296$	−26.9875	−1.56862
$297$	0	0
$298$	36.7845	2.13087
$299$	−4.25969	−0.246344
$300$	0	0
$301$	−1.33203	−0.0767772
$302$	28.5701	1.64403
$303$	0	0
$304$	16.1060	0.923740
$305$	−0.847929	−0.0485523
$306$	0	0
$307$	25.3593	1.44733	0.723667	−	0.690149i	$-0.242455\pi$
$307$	25.3593	1.44733	0.723667	+	0.690149i	$0.242455\pi$
$308$	−6.53592	−0.372419
$309$	0	0
$310$	31.0737	1.76487
$311$	−4.89777	−0.277727	−0.138864	−	0.990312i	$-0.544345\pi$
$311$	−4.89777	−0.277727	−0.138864	+	0.990312i	$0.544345\pi$
$312$	0	0
$313$	−5.31267	−0.300290	−0.150145	−	0.988664i	$-0.547974\pi$
$313$	−5.31267	−0.300290	−0.150145	+	0.988664i	$0.547974\pi$
$314$	−42.8054	−2.41565
$315$	0	0
$316$	−5.37195	−0.302196
$317$	15.5568	0.873755	0.436877	−	0.899521i	$-0.356084\pi$
$317$	15.5568	0.873755	0.436877	+	0.899521i	$0.356084\pi$
$318$	0	0
$319$	11.4946	0.643574
$320$	18.7313	1.04711
$321$	0	0
$322$	−18.8817	−1.05224
$323$	6.01181	0.334506
$324$	0	0
$325$	3.49879	0.194078
$326$	3.04860	0.168847
$327$	0	0
$328$	−35.9998	−1.98776
$329$	2.63698	0.145381
$330$	0	0
$331$	34.1918	1.87935	0.939677	−	0.342064i	$-0.111126\pi$
$331$	34.1918	1.87935	0.939677	+	0.342064i	$0.111126\pi$
$332$	−40.1686	−2.20454
$333$	0	0
$334$	20.6440	1.12959
$335$	6.34875	0.346869
$336$	0	0
$337$	30.3914	1.65552	0.827761	−	0.561080i	$-0.189614\pi$
$337$	30.3914	1.65552	0.827761	+	0.561080i	$0.189614\pi$
$338$	−31.5336	−1.71520
$339$	0	0
$340$	−26.7087	−1.44848
$341$	5.83149	0.315793
$342$	0	0
$343$	1.00000	0.0539949
$344$	−7.21560	−0.389039
$345$	0	0
$346$	14.9612	0.804319
$347$	1.66425	0.0893418	0.0446709	−	0.999002i	$-0.485776\pi$
$347$	1.66425	0.0893418	0.0446709	+	0.999002i	$0.485776\pi$
$348$	0	0
$349$	12.8777	0.689327	0.344663	−	0.938726i	$-0.387993\pi$
$349$	12.8777	0.689327	0.344663	+	0.938726i	$0.387993\pi$
$350$	15.5089	0.828987
$351$	0	0
$352$	−2.91106	−0.155160
$353$	13.5275	0.719997	0.359999	−	0.932953i	$-0.382777\pi$
$353$	13.5275	0.719997	0.359999	+	0.932953i	$0.382777\pi$
$354$	0	0
$355$	26.3120	1.39650
$356$	4.27020	0.226320
$357$	0	0
$358$	−50.2283	−2.65465
$359$	19.5330	1.03091	0.515457	−	0.856915i	$-0.327622\pi$
$359$	19.5330	1.03091	0.515457	+	0.856915i	$0.327622\pi$
$360$	0	0
$361$	−9.05477	−0.476567
$362$	−31.8060	−1.67169
$363$	0	0
$364$	2.34365	0.122841
$365$	−34.6394	−1.81311
$366$	0	0
$367$	−26.2079	−1.36804	−0.684022	−	0.729462i	$-0.739771\pi$
$367$	−26.2079	−1.36804	−0.684022	+	0.729462i	$0.739771\pi$
$368$	−38.7931	−2.02223
$369$	0	0
$370$	41.5178	2.15841
$371$	2.87115	0.149063
$372$	0	0
$373$	−32.8179	−1.69925	−0.849624	−	0.527388i	$-0.823171\pi$
$373$	−32.8179	−1.69925	−0.849624	+	0.527388i	$0.823171\pi$
$374$	−7.41104	−0.383216
$375$	0	0
$376$	14.2844	0.736664
$377$	−4.12173	−0.212280
$378$	0	0
$379$	15.8173	0.812481	0.406240	−	0.913766i	$-0.366840\pi$
$379$	15.8173	0.812481	0.406240	+	0.913766i	$0.366840\pi$
$380$	−44.1837	−2.26658
$381$	0	0
$382$	−40.3490	−2.06443
$383$	−15.5071	−0.792374	−0.396187	−	0.918170i	$-0.629667\pi$
$383$	−15.5071	−0.792374	−0.396187	+	0.918170i	$0.629667\pi$
$384$	0	0
$385$	5.24301	0.267209
$386$	−60.3290	−3.07067
$387$	0	0
$388$	−4.02848	−0.204515
$389$	−6.99370	−0.354595	−0.177297	−	0.984157i	$-0.556735\pi$
$389$	−6.99370	−0.354595	−0.177297	+	0.984157i	$0.556735\pi$
$390$	0	0
$391$	−14.4802	−0.732293
$392$	5.41698	0.273599
$393$	0	0
$394$	−16.3769	−0.825057
$395$	4.30929	0.216824
$396$	0	0
$397$	17.8751	0.897128	0.448564	−	0.893751i	$-0.351936\pi$
$397$	17.8751	0.897128	0.448564	+	0.893751i	$0.351936\pi$
$398$	43.7302	2.19200
$399$	0	0
$400$	31.8636	1.59318
$401$	19.0867	0.953143	0.476571	−	0.879136i	$-0.341879\pi$
$401$	19.0867	0.953143	0.476571	+	0.879136i	$0.341879\pi$
$402$	0	0
$403$	−2.09105	−0.104163
$404$	−47.6516	−2.37076
$405$	0	0
$406$	−18.2702	−0.906733
$407$	7.79151	0.386211
$408$	0	0
$409$	19.6481	0.971535	0.485768	−	0.874088i	$-0.338540\pi$
$409$	19.6481	0.971535	0.485768	+	0.874088i	$0.338540\pi$
$410$	55.3825	2.73515
$411$	0	0
$412$	−73.1543	−3.60405
$413$	11.5527	0.568472
$414$	0	0
$415$	32.2225	1.58174
$416$	1.04385	0.0511787
$417$	0	0
$418$	−12.2600	−0.599654
$419$	2.20594	0.107767	0.0538837	−	0.998547i	$-0.482840\pi$
$419$	2.20594	0.107767	0.0538837	+	0.998547i	$0.482840\pi$
$420$	0	0
$421$	−37.6865	−1.83673	−0.918363	−	0.395738i	$-0.870489\pi$
$421$	−37.6865	−1.83673	−0.918363	+	0.395738i	$0.870489\pi$
$422$	−22.8670	−1.11315
$423$	0	0
$424$	15.5530	0.755319
$425$	11.8936	0.576925
$426$	0	0
$427$	0.252927	0.0122400
$428$	25.3823	1.22690
$429$	0	0
$430$	11.1006	0.535316
$431$	41.2393	1.98643	0.993215	−	0.116297i	$-0.0371023\pi$
$431$	41.2393	1.98643	0.993215	+	0.116297i	$0.0371023\pi$
$432$	0	0
$433$	5.90179	0.283622	0.141811	−	0.989894i	$-0.454707\pi$
$433$	5.90179	0.283622	0.141811	+	0.989894i	$0.454707\pi$
$434$	−9.26890	−0.444921
$435$	0	0
$436$	−22.2539	−1.06577
$437$	−23.9543	−1.14589
$438$	0	0
$439$	18.1597	0.866716	0.433358	−	0.901222i	$-0.357329\pi$
$439$	18.1597	0.866716	0.433358	+	0.901222i	$0.357329\pi$
$440$	28.4013	1.35398
$441$	0	0
$442$	2.65745	0.126402
$443$	24.4277	1.16060	0.580298	−	0.814404i	$-0.302936\pi$
$443$	24.4277	1.16060	0.580298	+	0.814404i	$0.302936\pi$
$444$	0	0
$445$	−3.42548	−0.162383
$446$	−29.9351	−1.41747
$447$	0	0
$448$	−5.58731	−0.263976
$449$	−39.1985	−1.84989	−0.924946	−	0.380098i	$-0.875890\pi$
$449$	−39.1985	−1.84989	−0.924946	+	0.380098i	$0.875890\pi$
$450$	0	0
$451$	10.3934	0.489408
$452$	−68.7202	−3.23233
$453$	0	0
$454$	−5.64591	−0.264976
$455$	−1.88004	−0.0881374
$456$	0	0
$457$	−5.85512	−0.273891	−0.136945	−	0.990579i	$-0.543729\pi$
$457$	−5.85512	−0.273891	−0.136945	+	0.990579i	$0.543729\pi$
$458$	28.9400	1.35228
$459$	0	0
$460$	106.422	4.96194
$461$	−2.53290	−0.117969	−0.0589845	−	0.998259i	$-0.518786\pi$
$461$	−2.53290	−0.117969	−0.0589845	+	0.998259i	$0.518786\pi$
$462$	0	0
$463$	35.6610	1.65731	0.828653	−	0.559763i	$-0.189108\pi$
$463$	35.6610	1.65731	0.828653	+	0.559763i	$0.189108\pi$
$464$	−37.5367	−1.74260
$465$	0	0
$466$	30.2292	1.40034
$467$	40.7615	1.88622	0.943110	−	0.332482i	$-0.107886\pi$
$467$	40.7615	1.88622	0.943110	+	0.332482i	$0.107886\pi$
$468$	0	0
$469$	−1.89376	−0.0874456
$470$	−21.9753	−1.01365
$471$	0	0
$472$	62.5808	2.88052
$473$	2.08320	0.0957858
$474$	0	0
$475$	19.6754	0.902770
$476$	7.96688	0.365161
$477$	0	0
$478$	−19.7736	−0.904423
$479$	30.8787	1.41088	0.705441	−	0.708769i	$-0.250749\pi$
$479$	30.8787	1.41088	0.705441	+	0.708769i	$0.250749\pi$
$480$	0	0
$481$	−2.79387	−0.127390
$482$	24.3557	1.10937
$483$	0	0
$484$	−35.7492	−1.62496
$485$	3.23158	0.146738
$486$	0	0
$487$	−14.0096	−0.634836	−0.317418	−	0.948286i	$-0.602816\pi$
$487$	−14.0096	−0.634836	−0.317418	+	0.948286i	$0.602816\pi$
$488$	1.37010	0.0620216
$489$	0	0
$490$	−8.33354	−0.376471
$491$	21.2640	0.959630	0.479815	−	0.877370i	$-0.340704\pi$
$491$	21.2640	0.959630	0.479815	+	0.877370i	$0.340704\pi$
$492$	0	0
$493$	−14.0112	−0.631032
$494$	4.39617	0.197793
$495$	0	0
$496$	−19.0433	−0.855069
$497$	−7.84857	−0.352056
$498$	0	0
$499$	11.4598	0.513013	0.256506	−	0.966543i	$-0.417429\pi$
$499$	11.4598	0.513013	0.256506	+	0.966543i	$0.417429\pi$
$500$	−17.3593	−0.776332
$501$	0	0
$502$	−39.0661	−1.74360
$503$	16.6579	0.742741	0.371371	−	0.928485i	$-0.378888\pi$
$503$	16.6579	0.742741	0.371371	+	0.928485i	$0.378888\pi$
$504$	0	0
$505$	38.2253	1.70101
$506$	29.5296	1.31275
$507$	0	0
$508$	−4.17918	−0.185421
$509$	29.7651	1.31932	0.659658	−	0.751566i	$-0.270701\pi$
$509$	29.7651	1.31932	0.659658	+	0.751566i	$0.270701\pi$
$510$	0	0
$511$	10.3325	0.457083
$512$	−45.8244	−2.02517
$513$	0	0
$514$	50.7503	2.23850
$515$	58.6831	2.58589
$516$	0	0
$517$	−4.12404	−0.181375
$518$	−12.3843	−0.544133
$519$	0	0
$520$	−10.1841	−0.446603
$521$	42.7616	1.87342	0.936709	−	0.350108i	$-0.113855\pi$
$521$	42.7616	1.87342	0.936709	+	0.350108i	$0.113855\pi$
$522$	0	0
$523$	−21.4721	−0.938911	−0.469456	−	0.882956i	$-0.655550\pi$
$523$	−21.4721	−0.938911	−0.469456	+	0.882956i	$0.655550\pi$
$524$	−49.1151	−2.14560
$525$	0	0
$526$	−28.5384	−1.24433
$527$	−7.10821	−0.309639
$528$	0	0
$529$	34.6967	1.50855
$530$	−23.9268	−1.03932
$531$	0	0
$532$	13.1795	0.571403
$533$	−3.72688	−0.161429
$534$	0	0
$535$	−20.3613	−0.880295
$536$	−10.2584	−0.443098
$537$	0	0
$538$	−16.2367	−0.700014
$539$	−1.56393	−0.0673631
$540$	0	0
$541$	29.7590	1.27944	0.639720	−	0.768608i	$-0.279050\pi$
$541$	29.7590	1.27944	0.639720	+	0.768608i	$0.279050\pi$
$542$	−41.4893	−1.78212
$543$	0	0
$544$	3.54839	0.152136
$545$	17.8517	0.764683
$546$	0	0
$547$	37.4972	1.60326	0.801631	−	0.597819i	$-0.203966\pi$
$547$	37.4972	1.60326	0.801631	+	0.597819i	$0.203966\pi$
$548$	57.2901	2.44731
$549$	0	0
$550$	−24.2548	−1.03423
$551$	−23.1785	−0.987436
$552$	0	0
$553$	−1.28541	−0.0546612
$554$	−37.9788	−1.61356
$555$	0	0
$556$	77.5868	3.29041
$557$	15.0550	0.637902	0.318951	−	0.947771i	$-0.396669\pi$
$557$	15.0550	0.637902	0.318951	+	0.947771i	$0.396669\pi$
$558$	0	0
$559$	−0.746994	−0.0315945
$560$	−17.1216	−0.723518
$561$	0	0
$562$	47.6298	2.00914
$563$	39.8707	1.68035	0.840176	−	0.542315i	$-0.182452\pi$
$563$	39.8707	1.68035	0.840176	+	0.542315i	$0.182452\pi$
$564$	0	0
$565$	55.1262	2.31918
$566$	−65.7472	−2.76356
$567$	0	0
$568$	−42.5156	−1.78391
$569$	15.6121	0.654494	0.327247	−	0.944939i	$-0.393879\pi$
$569$	15.6121	0.654494	0.327247	+	0.944939i	$0.393879\pi$
$570$	0	0
$571$	−20.1708	−0.844122	−0.422061	−	0.906567i	$-0.638693\pi$
$571$	−20.1708	−0.844122	−0.422061	+	0.906567i	$0.638693\pi$
$572$	−3.66530	−0.153254
$573$	0	0
$574$	−16.5199	−0.689529
$575$	−47.3906	−1.97632
$576$	0	0
$577$	−2.60463	−0.108432	−0.0542160	−	0.998529i	$-0.517266\pi$
$577$	−2.60463	−0.108432	−0.0542160	+	0.998529i	$0.517266\pi$
$578$	−33.2249	−1.38197
$579$	0	0
$580$	102.975	4.27581
$581$	−9.61160	−0.398756
$582$	0	0
$583$	−4.49027	−0.185968
$584$	55.9710	2.31610
$585$	0	0
$586$	54.4216	2.24813
$587$	−3.91484	−0.161583	−0.0807915	−	0.996731i	$-0.525745\pi$
$587$	−3.91484	−0.161583	−0.0807915	+	0.996731i	$0.525745\pi$
$588$	0	0
$589$	−11.7590	−0.484521
$590$	−96.2749	−3.96358
$591$	0	0
$592$	−25.4439	−1.04574
$593$	12.3640	0.507729	0.253865	−	0.967240i	$-0.418298\pi$
$593$	12.3640	0.507729	0.253865	+	0.967240i	$0.418298\pi$
$594$	0	0
$595$	−6.39090	−0.262001
$596$	61.8430	2.53319
$597$	0	0
$598$	−10.5887	−0.433004
$599$	−5.47290	−0.223616	−0.111808	−	0.993730i	$-0.535664\pi$
$599$	−5.47290	−0.223616	−0.111808	+	0.993730i	$0.535664\pi$
$600$	0	0
$601$	−48.3519	−1.97232	−0.986158	−	0.165810i	$-0.946976\pi$
$601$	−48.3519	−1.97232	−0.986158	+	0.165810i	$0.946976\pi$
$602$	−3.31116	−0.134953
$603$	0	0
$604$	48.0328	1.95443
$605$	28.6774	1.16590
$606$	0	0
$607$	−3.89921	−0.158264	−0.0791319	−	0.996864i	$-0.525215\pi$
$607$	−3.89921	−0.158264	−0.0791319	+	0.996864i	$0.525215\pi$
$608$	5.87005	0.238062
$609$	0	0
$610$	−2.10778	−0.0853414
$611$	1.47880	0.0598256
$612$	0	0
$613$	12.4010	0.500872	0.250436	−	0.968133i	$-0.419426\pi$
$613$	12.4010	0.500872	0.250436	+	0.968133i	$0.419426\pi$
$614$	63.0381	2.54401
$615$	0	0
$616$	−8.47177	−0.341337
$617$	−9.77656	−0.393589	−0.196795	−	0.980445i	$-0.563053\pi$
$617$	−9.77656	−0.393589	−0.196795	+	0.980445i	$0.563053\pi$
$618$	0	0
$619$	−28.9771	−1.16469	−0.582344	−	0.812942i	$-0.697864\pi$
$619$	−28.9771	−1.16469	−0.582344	+	0.812942i	$0.697864\pi$
$620$	52.2417	2.09808
$621$	0	0
$622$	−12.1749	−0.488167
$623$	1.02178	0.0409368
$624$	0	0
$625$	−17.2697	−0.690790
$626$	−13.2062	−0.527826
$627$	0	0
$628$	−71.9654	−2.87173
$629$	−9.49735	−0.378684
$630$	0	0
$631$	9.75643	0.388397	0.194199	−	0.980962i	$-0.437789\pi$
$631$	9.75643	0.388397	0.194199	+	0.980962i	$0.437789\pi$
$632$	−6.96304	−0.276975
$633$	0	0
$634$	38.6709	1.53582
$635$	3.35246	0.133038
$636$	0	0
$637$	0.560792	0.0222194
$638$	28.5732	1.13122
$639$	0	0
$640$	59.0425	2.33386
$641$	43.0884	1.70189	0.850944	−	0.525256i	$-0.176031\pi$
$641$	43.0884	1.70189	0.850944	+	0.525256i	$0.176031\pi$
$642$	0	0
$643$	−27.8194	−1.09709	−0.548545	−	0.836121i	$-0.684818\pi$
$643$	−27.8194	−1.09709	−0.548545	+	0.836121i	$0.684818\pi$
$644$	−31.7443	−1.25090
$645$	0	0
$646$	14.9441	0.587968
$647$	4.76600	0.187371	0.0936854	−	0.995602i	$-0.470135\pi$
$647$	4.76600	0.187371	0.0936854	+	0.995602i	$0.470135\pi$
$648$	0	0
$649$	−18.0676	−0.709215
$650$	8.69728	0.341135
$651$	0	0
$652$	5.12538	0.200726
$653$	−39.6544	−1.55180	−0.775898	−	0.630859i	$-0.782703\pi$
$653$	−39.6544	−1.55180	−0.775898	+	0.630859i	$0.782703\pi$
$654$	0	0
$655$	39.3993	1.53946
$656$	−33.9408	−1.32517
$657$	0	0
$658$	6.55498	0.255540
$659$	−33.0049	−1.28569	−0.642844	−	0.765997i	$-0.722246\pi$
$659$	−33.0049	−1.28569	−0.642844	+	0.765997i	$0.722246\pi$
$660$	0	0
$661$	−15.9798	−0.621542	−0.310771	−	0.950485i	$-0.600587\pi$
$661$	−15.9798	−0.621542	−0.310771	+	0.950485i	$0.600587\pi$
$662$	84.9939	3.30338
$663$	0	0
$664$	−52.0659	−2.02055
$665$	−10.5724	−0.409978
$666$	0	0
$667$	55.8281	2.16167
$668$	34.7072	1.34286
$669$	0	0
$670$	15.7817	0.609700
$671$	−0.395560	−0.0152704
$672$	0	0
$673$	10.3527	0.399066	0.199533	−	0.979891i	$-0.436057\pi$
$673$	10.3527	0.399066	0.199533	+	0.979891i	$0.436057\pi$
$674$	75.5467	2.90995
$675$	0	0
$676$	−53.0150	−2.03904
$677$	35.7986	1.37585	0.687926	−	0.725781i	$-0.258521\pi$
$677$	35.7986	1.37585	0.687926	+	0.725781i	$0.258521\pi$
$678$	0	0
$679$	−0.963940	−0.0369926
$680$	−34.6194	−1.32759
$681$	0	0
$682$	14.4959	0.555076
$683$	−28.0300	−1.07254	−0.536268	−	0.844047i	$-0.680167\pi$
$683$	−28.0300	−1.07254	−0.536268	+	0.844047i	$0.680167\pi$
$684$	0	0
$685$	−45.9572	−1.75593
$686$	2.48579	0.0949081
$687$	0	0
$688$	−6.80291	−0.259358
$689$	1.61012	0.0613406
$690$	0	0
$691$	17.1163	0.651136	0.325568	−	0.945519i	$-0.394444\pi$
$691$	17.1163	0.651136	0.325568	+	0.945519i	$0.394444\pi$
$692$	25.1531	0.956177
$693$	0	0
$694$	4.13699	0.157038
$695$	−62.2388	−2.36085
$696$	0	0
$697$	−12.6690	−0.479871
$698$	32.0113	1.21165
$699$	0	0
$700$	26.0740	0.985503
$701$	26.1867	0.989057	0.494529	−	0.869161i	$-0.335341\pi$
$701$	26.1867	0.989057	0.494529	+	0.869161i	$0.335341\pi$
$702$	0	0
$703$	−15.7113	−0.592563
$704$	8.73815	0.329331
$705$	0	0
$706$	33.6266	1.26556
$707$	−11.4022	−0.428822
$708$	0	0
$709$	−11.5413	−0.433442	−0.216721	−	0.976234i	$-0.569536\pi$
$709$	−11.5413	−0.433442	−0.216721	+	0.976234i	$0.569536\pi$
$710$	65.4063	2.45466
$711$	0	0
$712$	5.53497	0.207432
$713$	28.3230	1.06070
$714$	0	0
$715$	2.94024	0.109959
$716$	−84.4449	−3.15585
$717$	0	0
$718$	48.5551	1.81206
$719$	−1.67651	−0.0625231	−0.0312615	−	0.999511i	$-0.509952\pi$
$719$	−1.67651	−0.0625231	−0.0312615	+	0.999511i	$0.509952\pi$
$720$	0	0
$721$	−17.5045	−0.651901
$722$	−22.5083	−0.837672
$723$	0	0
$724$	−53.4730	−1.98731
$725$	−45.8557	−1.70304
$726$	0	0
$727$	−37.3667	−1.38585	−0.692927	−	0.721007i	$-0.743679\pi$
$727$	−37.3667	−1.38585	−0.692927	+	0.721007i	$0.743679\pi$
$728$	3.03780	0.112588
$729$	0	0
$730$	−86.1063	−3.18694
$731$	−2.53929	−0.0939192
$732$	0	0
$733$	18.2078	0.672519	0.336259	−	0.941769i	$-0.390838\pi$
$733$	18.2078	0.672519	0.336259	+	0.941769i	$0.390838\pi$
$734$	−65.1476	−2.40464
$735$	0	0
$736$	−14.1387	−0.521160
$737$	2.96170	0.109096
$738$	0	0
$739$	34.4240	1.26631	0.633153	−	0.774027i	$-0.281760\pi$
$739$	34.4240	1.26631	0.633153	+	0.774027i	$0.281760\pi$
$740$	69.8007	2.56592
$741$	0	0
$742$	7.13709	0.262011
$743$	−38.3673	−1.40756	−0.703779	−	0.710419i	$-0.748506\pi$
$743$	−38.3673	−1.40756	−0.703779	+	0.710419i	$0.748506\pi$
$744$	0	0
$745$	−49.6094	−1.81755
$746$	−81.5787	−2.98681
$747$	0	0
$748$	−12.4596	−0.455569
$749$	6.07352	0.221922
$750$	0	0
$751$	−36.1410	−1.31880	−0.659401	−	0.751791i	$-0.729190\pi$
$751$	−36.1410	−1.31880	−0.659401	+	0.751791i	$0.729190\pi$
$752$	13.4674	0.491107
$753$	0	0
$754$	−10.2458	−0.373129
$755$	−38.5311	−1.40229
$756$	0	0
$757$	14.6536	0.532594	0.266297	−	0.963891i	$-0.414200\pi$
$757$	14.6536	0.532594	0.266297	+	0.963891i	$0.414200\pi$
$758$	39.3186	1.42812
$759$	0	0
$760$	−57.2703	−2.07741
$761$	−0.148905	−0.00539779	−0.00269890	−	0.999996i	$-0.500859\pi$
$761$	−0.148905	−0.00539779	−0.00269890	+	0.999996i	$0.500859\pi$
$762$	0	0
$763$	−5.32495	−0.192776
$764$	−67.8357	−2.45421
$765$	0	0
$766$	−38.5474	−1.39277
$767$	6.47867	0.233931
$768$	0	0
$769$	12.0858	0.435824	0.217912	−	0.975968i	$-0.430075\pi$
$769$	12.0858	0.435824	0.217912	+	0.975968i	$0.430075\pi$
$770$	13.0330	0.469678
$771$	0	0
$772$	−101.427	−3.65042
$773$	−24.1533	−0.868734	−0.434367	−	0.900736i	$-0.643028\pi$
$773$	−24.1533	−0.868734	−0.434367	+	0.900736i	$0.643028\pi$
$774$	0	0
$775$	−23.2637	−0.835658
$776$	−5.22165	−0.187446
$777$	0	0
$778$	−17.3849	−0.623279
$779$	−20.9580	−0.750900
$780$	0	0
$781$	12.2746	0.439219
$782$	−35.9947	−1.28717
$783$	0	0
$784$	5.10716	0.182398
$785$	57.7294	2.06045
$786$	0	0
$787$	−27.6573	−0.985875	−0.492938	−	0.870065i	$-0.664077\pi$
$787$	−27.6573	−0.985875	−0.492938	+	0.870065i	$0.664077\pi$
$788$	−27.5332	−0.980831
$789$	0	0
$790$	10.7120	0.381116
$791$	−16.4435	−0.584663
$792$	0	0
$793$	0.141840	0.00503687
$794$	44.4339	1.57690
$795$	0	0
$796$	73.5202	2.60585
$797$	−24.9036	−0.882132	−0.441066	−	0.897475i	$-0.645400\pi$
$797$	−24.9036	−0.882132	−0.441066	+	0.897475i	$0.645400\pi$
$798$	0	0
$799$	5.02694	0.177840
$800$	11.6132	0.410587
$801$	0	0
$802$	47.4455	1.67536
$803$	−16.1593	−0.570249
$804$	0	0
$805$	25.4648	0.897515
$806$	−5.19793	−0.183089
$807$	0	0
$808$	−61.7653	−2.17290
$809$	−41.8035	−1.46973	−0.734867	−	0.678211i	$-0.762756\pi$
$809$	−41.8035	−1.46973	−0.734867	+	0.678211i	$0.762756\pi$
$810$	0	0
$811$	15.7021	0.551375	0.275688	−	0.961247i	$-0.411094\pi$
$811$	15.7021	0.551375	0.275688	+	0.961247i	$0.411094\pi$
$812$	−30.7162	−1.07793
$813$	0	0
$814$	19.3681	0.678851
$815$	−4.11150	−0.144019
$816$	0	0
$817$	−4.20071	−0.146964
$818$	48.8411	1.70769
$819$	0	0
$820$	93.1104	3.25155
$821$	−44.3132	−1.54654	−0.773271	−	0.634075i	$-0.781381\pi$
$821$	−44.3132	−1.54654	−0.773271	+	0.634075i	$0.781381\pi$
$822$	0	0
$823$	24.2151	0.844085	0.422042	−	0.906576i	$-0.361313\pi$
$823$	24.2151	0.844085	0.422042	+	0.906576i	$0.361313\pi$
$824$	−94.8215	−3.30326
$825$	0	0
$826$	28.7177	0.999215
$827$	19.1531	0.666018	0.333009	−	0.942924i	$-0.391936\pi$
$827$	19.1531	0.666018	0.333009	+	0.942924i	$0.391936\pi$
$828$	0	0
$829$	4.55255	0.158117	0.0790583	−	0.996870i	$-0.474809\pi$
$829$	4.55255	0.158117	0.0790583	+	0.996870i	$0.474809\pi$
$830$	80.0986	2.78026
$831$	0	0
$832$	−3.13332	−0.108628
$833$	1.90633	0.0660503
$834$	0	0
$835$	−27.8415	−0.963495
$836$	−20.6117	−0.712872
$837$	0	0
$838$	5.48352	0.189425
$839$	3.76876	0.130112	0.0650561	−	0.997882i	$-0.479277\pi$
$839$	3.76876	0.130112	0.0650561	+	0.997882i	$0.479277\pi$
$840$	0	0
$841$	25.0200	0.862759
$842$	−93.6808	−3.22846
$843$	0	0
$844$	−38.4445	−1.32332
$845$	42.5277	1.46300
$846$	0	0
$847$	−8.55413	−0.293923
$848$	14.6634	0.503543
$849$	0	0
$850$	29.5651	1.01407
$851$	37.8426	1.29723
$852$	0	0
$853$	35.2843	1.20811	0.604056	−	0.796942i	$-0.293550\pi$
$853$	35.2843	1.20811	0.604056	+	0.796942i	$0.293550\pi$
$854$	0.628725	0.0215145
$855$	0	0
$856$	32.9002	1.12450
$857$	−4.31950	−0.147551	−0.0737756	−	0.997275i	$-0.523505\pi$
$857$	−4.31950	−0.147551	−0.0737756	+	0.997275i	$0.523505\pi$
$858$	0	0
$859$	55.1455	1.88154	0.940771	−	0.339044i	$-0.110103\pi$
$859$	55.1455	1.88154	0.940771	+	0.339044i	$0.110103\pi$
$860$	18.6625	0.636386
$861$	0	0
$862$	102.513	3.49159
$863$	−9.45516	−0.321857	−0.160929	−	0.986966i	$-0.551449\pi$
$863$	−9.45516	−0.321857	−0.160929	+	0.986966i	$0.551449\pi$
$864$	0	0
$865$	−20.1774	−0.686052
$866$	14.6706	0.498529
$867$	0	0
$868$	−15.5831	−0.528924
$869$	2.01029	0.0681943
$870$	0	0
$871$	−1.06200	−0.0359846
$872$	−28.8452	−0.976820
$873$	0	0
$874$	−59.5454	−2.01415
$875$	−4.15377	−0.140423
$876$	0	0
$877$	−46.5998	−1.57356	−0.786782	−	0.617231i	$-0.788254\pi$
$877$	−46.5998	−1.57356	−0.786782	+	0.617231i	$0.788254\pi$
$878$	45.1413	1.52345
$879$	0	0
$880$	26.7769	0.902648
$881$	22.5999	0.761409	0.380705	−	0.924697i	$-0.375681\pi$
$881$	22.5999	0.761409	0.380705	+	0.924697i	$0.375681\pi$
$882$	0	0
$883$	−8.21563	−0.276478	−0.138239	−	0.990399i	$-0.544144\pi$
$883$	−8.21563	−0.276478	−0.138239	+	0.990399i	$0.544144\pi$
$884$	4.46776	0.150267
$885$	0	0
$886$	60.7223	2.04001
$887$	−39.4261	−1.32380	−0.661899	−	0.749593i	$-0.730249\pi$
$887$	−39.4261	−1.32380	−0.661899	+	0.749593i	$0.730249\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−8.51505	−0.285425
$891$	0	0
$892$	−50.3276	−1.68509
$893$	8.31598	0.278284
$894$	0	0
$895$	67.7403	2.26431
$896$	−17.6117	−0.588364
$897$	0	0
$898$	−97.4395	−3.25160
$899$	27.4057	0.914030
$900$	0	0
$901$	5.47335	0.182344
$902$	25.8360	0.860244
$903$	0	0
$904$	−89.0741	−2.96256
$905$	42.8952	1.42588
$906$	0	0
$907$	7.21394	0.239535	0.119767	−	0.992802i	$-0.461785\pi$
$907$	7.21394	0.239535	0.119767	+	0.992802i	$0.461785\pi$
$908$	−9.49203	−0.315004
$909$	0	0
$910$	−4.67338	−0.154921
$911$	3.62410	0.120072	0.0600360	−	0.998196i	$-0.480878\pi$
$911$	3.62410	0.120072	0.0600360	+	0.998196i	$0.480878\pi$
$912$	0	0
$913$	15.0318	0.497481
$914$	−14.5546	−0.481424
$915$	0	0
$916$	48.6547	1.60760
$917$	−11.7524	−0.388097
$918$	0	0
$919$	12.3836	0.408498	0.204249	−	0.978919i	$-0.434525\pi$
$919$	12.3836	0.408498	0.204249	+	0.978919i	$0.434525\pi$
$920$	137.942	4.54782
$921$	0	0
$922$	−6.29627	−0.207357
$923$	−4.40142	−0.144874
$924$	0	0
$925$	−31.0829	−1.02200
$926$	88.6458	2.91308
$927$	0	0
$928$	−13.6808	−0.449094
$929$	−56.3033	−1.84725	−0.923625	−	0.383296i	$-0.874789\pi$
$929$	−56.3033	−1.84725	−0.923625	+	0.383296i	$0.874789\pi$
$930$	0	0
$931$	3.15361	0.103355
$932$	50.8221	1.66473
$933$	0	0
$934$	101.325	3.31545
$935$	9.99489	0.326868
$936$	0	0
$937$	−14.5952	−0.476805	−0.238403	−	0.971166i	$-0.576624\pi$
$937$	−14.5952	−0.476805	−0.238403	+	0.971166i	$0.576624\pi$
$938$	−4.70749	−0.153705
$939$	0	0
$940$	−36.9454	−1.20503
$941$	−59.0179	−1.92393	−0.961965	−	0.273173i	$-0.911927\pi$
$941$	−59.0179	−1.92393	−0.961965	+	0.273173i	$0.911927\pi$
$942$	0	0
$943$	50.4799	1.64385
$944$	59.0015	1.92033
$945$	0	0
$946$	5.17842	0.168365
$947$	−7.29868	−0.237175	−0.118588	−	0.992944i	$-0.537837\pi$
$947$	−7.29868	−0.237175	−0.118588	+	0.992944i	$0.537837\pi$
$948$	0	0
$949$	5.79439	0.188094
$950$	48.9090	1.58682
$951$	0	0
$952$	10.3265	0.334685
$953$	−20.8825	−0.676449	−0.338225	−	0.941065i	$-0.609826\pi$
$953$	−20.8825	−0.676449	−0.338225	+	0.941065i	$0.609826\pi$
$954$	0	0
$955$	54.4166	1.76088
$956$	−33.2438	−1.07518
$957$	0	0
$958$	76.7580	2.47994
$959$	13.7085	0.442670
$960$	0	0
$961$	−17.0964	−0.551498
$962$	−6.94500	−0.223916
$963$	0	0
$964$	40.9474	1.31883
$965$	81.3627	2.61916
$966$	0	0
$967$	−8.16712	−0.262637	−0.131318	−	0.991340i	$-0.541921\pi$
$967$	−8.16712	−0.262637	−0.131318	+	0.991340i	$0.541921\pi$
$968$	−46.3376	−1.48935
$969$	0	0
$970$	8.03303	0.257925
$971$	21.7639	0.698438	0.349219	−	0.937041i	$-0.386447\pi$
$971$	21.7639	0.698438	0.349219	+	0.937041i	$0.386447\pi$
$972$	0	0
$973$	18.5651	0.595170
$974$	−34.8250	−1.11586
$975$	0	0
$976$	1.29174	0.0413475
$977$	−26.9152	−0.861093	−0.430547	−	0.902568i	$-0.641679\pi$
$977$	−26.9152	−0.861093	−0.430547	+	0.902568i	$0.641679\pi$
$978$	0	0
$979$	−1.59799	−0.0510720
$980$	−14.0105	−0.447550
$981$	0	0
$982$	52.8579	1.68676
$983$	−25.9287	−0.826997	−0.413499	−	0.910505i	$-0.635693\pi$
$983$	−25.9287	−0.826997	−0.413499	+	0.910505i	$0.635693\pi$
$984$	0	0
$985$	22.0867	0.703741
$986$	−34.8289	−1.10918
$987$	0	0
$988$	7.39095	0.235137
$989$	10.1179	0.321731
$990$	0	0
$991$	−16.9040	−0.536972	−0.268486	−	0.963284i	$-0.586523\pi$
$991$	−16.9040	−0.536972	−0.268486	+	0.963284i	$0.586523\pi$
$992$	−6.94060	−0.220364
$993$	0	0
$994$	−19.5099	−0.618817
$995$	−58.9767	−1.86969
$996$	0	0
$997$	−19.5107	−0.617909	−0.308955	−	0.951077i	$-0.599979\pi$
$997$	−19.5107	−0.617909	−0.308955	+	0.951077i	$0.599979\pi$
$998$	28.4868	0.901734
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.20		20
3.2	odd	2		889.2.a.d.1.1	✓	20
21.20	even	2		6223.2.a.l.1.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.1	✓	20	3.2	odd	2
6223.2.a.l.1.1		20	21.20	even	2
8001.2.a.w.1.20		20	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)

\(f(q)\)	\(=\)	\(q+2.48579 q^{2} +4.17918 q^{4} -3.35246 q^{5} +1.00000 q^{7} +5.41698 q^{8} +O(q^{10})\)	\(q+2.48579 q^{2} +4.17918 q^{4} -3.35246 q^{5} +1.00000 q^{7} +5.41698 q^{8} -8.33354 q^{10} -1.56393 q^{11} +0.560792 q^{13} +2.48579 q^{14} +5.10716 q^{16} +1.90633 q^{17} +3.15361 q^{19} -14.0105 q^{20} -3.88760 q^{22} -7.59584 q^{23} +6.23902 q^{25} +1.39401 q^{26} +4.17918 q^{28} -7.34983 q^{29} -3.72875 q^{31} +1.86138 q^{32} +4.73874 q^{34} -3.35246 q^{35} -4.98201 q^{37} +7.83922 q^{38} -18.1602 q^{40} -6.64574 q^{41} -1.33203 q^{43} -6.53592 q^{44} -18.8817 q^{46} +2.63698 q^{47} +1.00000 q^{49} +15.5089 q^{50} +2.34365 q^{52} +2.87115 q^{53} +5.24301 q^{55} +5.41698 q^{56} -18.2702 q^{58} +11.5527 q^{59} +0.252927 q^{61} -9.26890 q^{62} -5.58731 q^{64} -1.88004 q^{65} -1.89376 q^{67} +7.96688 q^{68} -8.33354 q^{70} -7.84857 q^{71} +10.3325 q^{73} -12.3843 q^{74} +13.1795 q^{76} -1.56393 q^{77} -1.28541 q^{79} -17.1216 q^{80} -16.5199 q^{82} -9.61160 q^{83} -6.39090 q^{85} -3.31116 q^{86} -8.47177 q^{88} +1.02178 q^{89} +0.560792 q^{91} -31.7443 q^{92} +6.55498 q^{94} -10.5724 q^{95} -0.963940 q^{97} +2.48579 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

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Database

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