LMFDB - Embedded newform 8001.2.a.w.1.19

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			1.19
Root			$-2.12365$ 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)

$f(q)$	$=$	$q+2.12365 q^{2} +2.50989 q^{4} +2.22935 q^{5} +1.00000 q^{7} +1.08284 q^{8} +O(q^{10})$	\(q+2.12365 q^{2} +2.50989 q^{4} +2.22935 q^{5} +1.00000 q^{7} +1.08284 q^{8} +4.73436 q^{10} -3.48053 q^{11} -5.41214 q^{13} +2.12365 q^{14} -2.72022 q^{16} -2.65180 q^{17} -5.06567 q^{19} +5.59544 q^{20} -7.39142 q^{22} +3.41536 q^{23} -0.0299934 q^{25} -11.4935 q^{26} +2.50989 q^{28} +0.715374 q^{29} +6.85305 q^{31} -7.94247 q^{32} -5.63149 q^{34} +2.22935 q^{35} -7.20777 q^{37} -10.7577 q^{38} +2.41403 q^{40} -3.98381 q^{41} -4.05383 q^{43} -8.73575 q^{44} +7.25303 q^{46} -3.64799 q^{47} +1.00000 q^{49} -0.0636955 q^{50} -13.5839 q^{52} +0.167103 q^{53} -7.75931 q^{55} +1.08284 q^{56} +1.51920 q^{58} +12.1971 q^{59} -3.87632 q^{61} +14.5535 q^{62} -11.4266 q^{64} -12.0656 q^{65} +14.7726 q^{67} -6.65573 q^{68} +4.73436 q^{70} -7.23109 q^{71} -7.89631 q^{73} -15.3068 q^{74} -12.7143 q^{76} -3.48053 q^{77} -13.7201 q^{79} -6.06432 q^{80} -8.46023 q^{82} -3.57015 q^{83} -5.91179 q^{85} -8.60892 q^{86} -3.76885 q^{88} +10.2897 q^{89} -5.41214 q^{91} +8.57219 q^{92} -7.74706 q^{94} -11.2931 q^{95} -14.1892 q^{97} +2.12365 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.12365	1.50165	0.750824	−	0.660502i	$-0.229657\pi$
$2$	2.12365	1.50165	0.750824	+	0.660502i	$0.229657\pi$
$3$	0	0
$3$	0	0
$4$	2.50989	1.25495
$5$	2.22935	0.996996	0.498498	−	0.866891i	$-0.333885\pi$
$5$	2.22935	0.996996	0.498498	+	0.866891i	$0.333885\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.08284	0.382841
$9$	0	0
$10$	4.73436	1.49714
$11$	−3.48053	−1.04942	−0.524709	−	0.851282i	$-0.675826\pi$
$11$	−3.48053	−1.04942	−0.524709	+	0.851282i	$0.675826\pi$
$12$	0	0
$13$	−5.41214	−1.50106	−0.750529	−	0.660837i	$-0.770201\pi$
$13$	−5.41214	−1.50106	−0.750529	+	0.660837i	$0.770201\pi$
$14$	2.12365	0.567570
$15$	0	0
$16$	−2.72022	−0.680055
$17$	−2.65180	−0.643155	−0.321578	−	0.946883i	$-0.604213\pi$
$17$	−2.65180	−0.643155	−0.321578	+	0.946883i	$0.604213\pi$
$18$	0	0
$19$	−5.06567	−1.16214	−0.581072	−	0.813852i	$-0.697366\pi$
$19$	−5.06567	−1.16214	−0.581072	+	0.813852i	$0.697366\pi$
$20$	5.59544	1.25118
$21$	0	0
$22$	−7.39142	−1.57586
$23$	3.41536	0.712152	0.356076	−	0.934457i	$-0.384114\pi$
$23$	3.41536	0.712152	0.356076	+	0.934457i	$0.384114\pi$
$24$	0	0
$25$	−0.0299934	−0.00599868
$26$	−11.4935	−2.25406
$27$	0	0
$28$	2.50989	0.474325
$29$	0.715374	0.132842	0.0664208	−	0.997792i	$-0.478842\pi$
$29$	0.715374	0.132842	0.0664208	+	0.997792i	$0.478842\pi$
$30$	0	0
$31$	6.85305	1.23084	0.615422	−	0.788198i	$-0.288986\pi$
$31$	6.85305	1.23084	0.615422	+	0.788198i	$0.288986\pi$
$32$	−7.94247	−1.40404
$33$	0	0
$34$	−5.63149	−0.965793
$35$	2.22935	0.376829
$36$	0	0
$37$	−7.20777	−1.18495	−0.592475	−	0.805589i	$-0.701849\pi$
$37$	−7.20777	−1.18495	−0.592475	+	0.805589i	$0.701849\pi$
$38$	−10.7577	−1.74513
$39$	0	0
$40$	2.41403	0.381691
$41$	−3.98381	−0.622167	−0.311084	−	0.950383i	$-0.600692\pi$
$41$	−3.98381	−0.622167	−0.311084	+	0.950383i	$0.600692\pi$
$42$	0	0
$43$	−4.05383	−0.618203	−0.309102	−	0.951029i	$-0.600028\pi$
$43$	−4.05383	−0.618203	−0.309102	+	0.951029i	$0.600028\pi$
$44$	−8.73575	−1.31696
$45$	0	0
$46$	7.25303	1.06940
$47$	−3.64799	−0.532114	−0.266057	−	0.963957i	$-0.585721\pi$
$47$	−3.64799	−0.532114	−0.266057	+	0.963957i	$0.585721\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.0636955	−0.00900791
$51$	0	0
$52$	−13.5839	−1.88375
$53$	0.167103	0.0229534	0.0114767	−	0.999934i	$-0.496347\pi$
$53$	0.167103	0.0229534	0.0114767	+	0.999934i	$0.496347\pi$
$54$	0	0
$55$	−7.75931	−1.04627
$56$	1.08284	0.144700
$57$	0	0
$58$	1.51920	0.199481
$59$	12.1971	1.58792	0.793961	−	0.607969i	$-0.208016\pi$
$59$	12.1971	1.58792	0.793961	+	0.607969i	$0.208016\pi$
$60$	0	0
$61$	−3.87632	−0.496312	−0.248156	−	0.968720i	$-0.579825\pi$
$61$	−3.87632	−0.496312	−0.248156	+	0.968720i	$0.579825\pi$
$62$	14.5535	1.84829
$63$	0	0
$64$	−11.4266	−1.42833
$65$	−12.0656	−1.49655
$66$	0	0
$67$	14.7726	1.80476	0.902379	−	0.430943i	$-0.141819\pi$
$67$	14.7726	1.80476	0.902379	+	0.430943i	$0.141819\pi$
$68$	−6.65573	−0.807126
$69$	0	0
$70$	4.73436	0.565865
$71$	−7.23109	−0.858172	−0.429086	−	0.903264i	$-0.641164\pi$
$71$	−7.23109	−0.858172	−0.429086	+	0.903264i	$0.641164\pi$
$72$	0	0
$73$	−7.89631	−0.924193	−0.462097	−	0.886830i	$-0.652903\pi$
$73$	−7.89631	−0.924193	−0.462097	+	0.886830i	$0.652903\pi$
$74$	−15.3068	−1.77938
$75$	0	0
$76$	−12.7143	−1.45843
$77$	−3.48053	−0.396643
$78$	0	0
$79$	−13.7201	−1.54363	−0.771815	−	0.635847i	$-0.780651\pi$
$79$	−13.7201	−1.54363	−0.771815	+	0.635847i	$0.780651\pi$
$80$	−6.06432	−0.678012
$81$	0	0
$82$	−8.46023	−0.934276
$83$	−3.57015	−0.391875	−0.195938	−	0.980616i	$-0.562775\pi$
$83$	−3.57015	−0.391875	−0.195938	+	0.980616i	$0.562775\pi$
$84$	0	0
$85$	−5.91179	−0.641223
$86$	−8.60892	−0.928324
$87$	0	0
$88$	−3.76885	−0.401760
$89$	10.2897	1.09070	0.545351	−	0.838208i	$-0.316396\pi$
$89$	10.2897	1.09070	0.545351	+	0.838208i	$0.316396\pi$
$90$	0	0
$91$	−5.41214	−0.567347
$92$	8.57219	0.893713
$93$	0	0
$94$	−7.74706	−0.799048
$95$	−11.2931	−1.15865
$96$	0	0
$97$	−14.1892	−1.44070	−0.720349	−	0.693612i	$-0.756018\pi$
$97$	−14.1892	−1.44070	−0.720349	+	0.693612i	$0.756018\pi$
$98$	2.12365	0.214521
$99$	0	0
$100$	−0.0752803	−0.00752803
$101$	−11.1299	−1.10746	−0.553732	−	0.832695i	$-0.686797\pi$
$101$	−11.1299	−1.10746	−0.553732	+	0.832695i	$0.686797\pi$
$102$	0	0
$103$	17.5185	1.72614	0.863072	−	0.505080i	$-0.168537\pi$
$103$	17.5185	1.72614	0.863072	+	0.505080i	$0.168537\pi$
$104$	−5.86048	−0.574667
$105$	0	0
$106$	0.354869	0.0344679
$107$	−7.19461	−0.695529	−0.347765	−	0.937582i	$-0.613059\pi$
$107$	−7.19461	−0.695529	−0.347765	+	0.937582i	$0.613059\pi$
$108$	0	0
$109$	1.82066	0.174387	0.0871936	−	0.996191i	$-0.472210\pi$
$109$	1.82066	0.174387	0.0871936	+	0.996191i	$0.472210\pi$
$110$	−16.4781	−1.57112
$111$	0	0
$112$	−2.72022	−0.257036
$113$	−7.44734	−0.700587	−0.350293	−	0.936640i	$-0.613918\pi$
$113$	−7.44734	−0.700587	−0.350293	+	0.936640i	$0.613918\pi$
$114$	0	0
$115$	7.61404	0.710013
$116$	1.79551	0.166709
$117$	0	0
$118$	25.9023	2.38450
$119$	−2.65180	−0.243090
$120$	0	0
$121$	1.11406	0.101278
$122$	−8.23195	−0.745286
$123$	0	0
$124$	17.2004	1.54464
$125$	−11.2136	−1.00298
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−8.38117	−0.740798
$129$	0	0
$130$	−25.6231	−2.24729
$131$	−17.6856	−1.54520	−0.772601	−	0.634892i	$-0.781045\pi$
$131$	−17.6856	−1.54520	−0.772601	+	0.634892i	$0.781045\pi$
$132$	0	0
$133$	−5.06567	−0.439249
$134$	31.3718	2.71011
$135$	0	0
$136$	−2.87147	−0.246226
$137$	13.9859	1.19490	0.597449	−	0.801907i	$-0.296181\pi$
$137$	13.9859	1.19490	0.597449	+	0.801907i	$0.296181\pi$
$138$	0	0
$139$	14.1006	1.19600	0.598000	−	0.801496i	$-0.295962\pi$
$139$	14.1006	1.19600	0.598000	+	0.801496i	$0.295962\pi$
$140$	5.59544	0.472901
$141$	0	0
$142$	−15.3563	−1.28867
$143$	18.8371	1.57524
$144$	0	0
$145$	1.59482	0.132443
$146$	−16.7690	−1.38781
$147$	0	0
$148$	−18.0908	−1.48705
$149$	9.05754	0.742022	0.371011	−	0.928628i	$-0.379011\pi$
$149$	9.05754	0.742022	0.371011	+	0.928628i	$0.379011\pi$
$150$	0	0
$151$	9.73199	0.791978	0.395989	−	0.918255i	$-0.370402\pi$
$151$	9.73199	0.791978	0.395989	+	0.918255i	$0.370402\pi$
$152$	−5.48530	−0.444916
$153$	0	0
$154$	−7.39142	−0.595618
$155$	15.2778	1.22715
$156$	0	0
$157$	20.6822	1.65062	0.825310	−	0.564680i	$-0.191000\pi$
$157$	20.6822	1.65062	0.825310	+	0.564680i	$0.191000\pi$
$158$	−29.1367	−2.31799
$159$	0	0
$160$	−17.7066	−1.39983
$161$	3.41536	0.269168
$162$	0	0
$163$	3.68850	0.288906	0.144453	−	0.989512i	$-0.453858\pi$
$163$	3.68850	0.288906	0.144453	+	0.989512i	$0.453858\pi$
$164$	−9.99895	−0.780787
$165$	0	0
$166$	−7.58176	−0.588459
$167$	−2.77511	−0.214744	−0.107372	−	0.994219i	$-0.534244\pi$
$167$	−2.77511	−0.214744	−0.107372	+	0.994219i	$0.534244\pi$
$168$	0	0
$169$	16.2913	1.25318
$170$	−12.5546	−0.962892
$171$	0	0
$172$	−10.1747	−0.775813
$173$	−4.61738	−0.351053	−0.175526	−	0.984475i	$-0.556163\pi$
$173$	−4.61738	−0.351053	−0.175526	+	0.984475i	$0.556163\pi$
$174$	0	0
$175$	−0.0299934	−0.00226729
$176$	9.46779	0.713662
$177$	0	0
$178$	21.8516	1.63785
$179$	2.08732	0.156014	0.0780068	−	0.996953i	$-0.475144\pi$
$179$	2.08732	0.156014	0.0780068	+	0.996953i	$0.475144\pi$
$180$	0	0
$181$	−11.0172	−0.818900	−0.409450	−	0.912333i	$-0.634279\pi$
$181$	−11.0172	−0.818900	−0.409450	+	0.912333i	$0.634279\pi$
$182$	−11.4935	−0.851955
$183$	0	0
$184$	3.69828	0.272641
$185$	−16.0687	−1.18139
$186$	0	0
$187$	9.22965	0.674939
$188$	−9.15607	−0.667775
$189$	0	0
$190$	−23.9827	−1.73989
$191$	−1.14009	−0.0824940	−0.0412470	−	0.999149i	$-0.513133\pi$
$191$	−1.14009	−0.0824940	−0.0412470	+	0.999149i	$0.513133\pi$
$192$	0	0
$193$	−17.2110	−1.23887	−0.619437	−	0.785047i	$-0.712639\pi$
$193$	−17.2110	−1.23887	−0.619437	+	0.785047i	$0.712639\pi$
$194$	−30.1330	−2.16342
$195$	0	0
$196$	2.50989	0.179278
$197$	3.20396	0.228273	0.114136	−	0.993465i	$-0.463590\pi$
$197$	3.20396	0.228273	0.114136	+	0.993465i	$0.463590\pi$
$198$	0	0
$199$	−13.8904	−0.984663	−0.492331	−	0.870408i	$-0.663855\pi$
$199$	−13.8904	−0.984663	−0.492331	+	0.870408i	$0.663855\pi$
$200$	−0.0324780	−0.00229654
$201$	0	0
$202$	−23.6360	−1.66302
$203$	0.715374	0.0502094
$204$	0	0
$205$	−8.88132	−0.620298
$206$	37.2031	2.59206
$207$	0	0
$208$	14.7222	1.02080
$209$	17.6312	1.21957
$210$	0	0
$211$	14.0063	0.964233	0.482117	−	0.876107i	$-0.339868\pi$
$211$	14.0063	0.964233	0.482117	+	0.876107i	$0.339868\pi$
$212$	0.419411	0.0288053
$213$	0	0
$214$	−15.2788	−1.04444
$215$	−9.03741	−0.616346
$216$	0	0
$217$	6.85305	0.465215
$218$	3.86644	0.261868
$219$	0	0
$220$	−19.4751	−1.31301
$221$	14.3519	0.965414
$222$	0	0
$223$	−5.79465	−0.388038	−0.194019	−	0.980998i	$-0.562152\pi$
$223$	−5.79465	−0.388038	−0.194019	+	0.980998i	$0.562152\pi$
$224$	−7.94247	−0.530679
$225$	0	0
$226$	−15.8156	−1.05203
$227$	0.299716	0.0198928	0.00994642	−	0.999951i	$-0.496834\pi$
$227$	0.299716	0.0198928	0.00994642	+	0.999951i	$0.496834\pi$
$228$	0	0
$229$	24.7530	1.63572	0.817862	−	0.575415i	$-0.195159\pi$
$229$	24.7530	1.63572	0.817862	+	0.575415i	$0.195159\pi$
$230$	16.1696	1.06619
$231$	0	0
$232$	0.774634	0.0508572
$233$	−6.52314	−0.427345	−0.213673	−	0.976905i	$-0.568543\pi$
$233$	−6.52314	−0.427345	−0.213673	+	0.976905i	$0.568543\pi$
$234$	0	0
$235$	−8.13265	−0.530516
$236$	30.6133	1.99276
$237$	0	0
$238$	−5.63149	−0.365035
$239$	−11.0844	−0.716991	−0.358496	−	0.933531i	$-0.616710\pi$
$239$	−11.0844	−0.716991	−0.358496	+	0.933531i	$0.616710\pi$
$240$	0	0
$241$	5.91832	0.381232	0.190616	−	0.981665i	$-0.438951\pi$
$241$	5.91832	0.381232	0.190616	+	0.981665i	$0.438951\pi$
$242$	2.36588	0.152084
$243$	0	0
$244$	−9.72915	−0.622845
$245$	2.22935	0.142428
$246$	0	0
$247$	27.4161	1.74445
$248$	7.42074	0.471217
$249$	0	0
$250$	−23.8138	−1.50612
$251$	−23.9356	−1.51080	−0.755399	−	0.655265i	$-0.772557\pi$
$251$	−23.9356	−1.51080	−0.755399	+	0.655265i	$0.772557\pi$
$252$	0	0
$253$	−11.8872	−0.747345
$254$	−2.12365	−0.133250
$255$	0	0
$256$	5.05451	0.315907
$257$	20.6473	1.28795	0.643973	−	0.765048i	$-0.277285\pi$
$257$	20.6473	1.28795	0.643973	+	0.765048i	$0.277285\pi$
$258$	0	0
$259$	−7.20777	−0.447869
$260$	−30.2833	−1.87809
$261$	0	0
$262$	−37.5581	−2.32035
$263$	−23.1113	−1.42511	−0.712553	−	0.701618i	$-0.752461\pi$
$263$	−23.1113	−1.42511	−0.712553	+	0.701618i	$0.752461\pi$
$264$	0	0
$265$	0.372531	0.0228844
$266$	−10.7577	−0.659597
$267$	0	0
$268$	37.0776	2.26488
$269$	17.0309	1.03839	0.519195	−	0.854656i	$-0.326232\pi$
$269$	17.0309	1.03839	0.519195	+	0.854656i	$0.326232\pi$
$270$	0	0
$271$	25.8143	1.56811	0.784054	−	0.620692i	$-0.213148\pi$
$271$	25.8143	1.56811	0.784054	+	0.620692i	$0.213148\pi$
$272$	7.21347	0.437381
$273$	0	0
$274$	29.7012	1.79432
$275$	0.104393	0.00629512
$276$	0	0
$277$	20.2565	1.21710	0.608548	−	0.793517i	$-0.291752\pi$
$277$	20.2565	1.21710	0.608548	+	0.793517i	$0.291752\pi$
$278$	29.9448	1.79597
$279$	0	0
$280$	2.41403	0.144266
$281$	12.0350	0.717945	0.358973	−	0.933348i	$-0.383127\pi$
$281$	12.0350	0.717945	0.358973	+	0.933348i	$0.383127\pi$
$282$	0	0
$283$	9.87709	0.587132	0.293566	−	0.955939i	$-0.405158\pi$
$283$	9.87709	0.587132	0.293566	+	0.955939i	$0.405158\pi$
$284$	−18.1493	−1.07696
$285$	0	0
$286$	40.0035	2.36545
$287$	−3.98381	−0.235157
$288$	0	0
$289$	−9.96797	−0.586351
$290$	3.38684	0.198882
$291$	0	0
$292$	−19.8189	−1.15981
$293$	−27.3421	−1.59734	−0.798670	−	0.601769i	$-0.794463\pi$
$293$	−27.3421	−1.59734	−0.798670	+	0.601769i	$0.794463\pi$
$294$	0	0
$295$	27.1915	1.58315
$296$	−7.80485	−0.453648
$297$	0	0
$298$	19.2351	1.11426
$299$	−18.4844	−1.06898
$300$	0	0
$301$	−4.05383	−0.233659
$302$	20.6674	1.18927
$303$	0	0
$304$	13.7797	0.790321
$305$	−8.64167	−0.494821
$306$	0	0
$307$	4.56385	0.260472	0.130236	−	0.991483i	$-0.458426\pi$
$307$	4.56385	0.260472	0.130236	+	0.991483i	$0.458426\pi$
$308$	−8.73575	−0.497766
$309$	0	0
$310$	32.4448	1.84274
$311$	−23.8213	−1.35078	−0.675390	−	0.737461i	$-0.736025\pi$
$311$	−23.8213	−1.35078	−0.675390	+	0.737461i	$0.736025\pi$
$312$	0	0
$313$	9.89656	0.559387	0.279693	−	0.960089i	$-0.409767\pi$
$313$	9.89656	0.559387	0.279693	+	0.960089i	$0.409767\pi$
$314$	43.9218	2.47865
$315$	0	0
$316$	−34.4360	−1.93717
$317$	−33.3827	−1.87496	−0.937481	−	0.348038i	$-0.886848\pi$
$317$	−33.3827	−1.87496	−0.937481	+	0.348038i	$0.886848\pi$
$318$	0	0
$319$	−2.48988	−0.139406
$320$	−25.4739	−1.42403
$321$	0	0
$322$	7.25303	0.404196
$323$	13.4331	0.747439
$324$	0	0
$325$	0.162329	0.00900437
$326$	7.83309	0.433835
$327$	0	0
$328$	−4.31383	−0.238191
$329$	−3.64799	−0.201120
$330$	0	0
$331$	−8.41323	−0.462433	−0.231217	−	0.972902i	$-0.574271\pi$
$331$	−8.41323	−0.462433	−0.231217	+	0.972902i	$0.574271\pi$
$332$	−8.96071	−0.491783
$333$	0	0
$334$	−5.89336	−0.322470
$335$	32.9333	1.79934
$336$	0	0
$337$	9.11846	0.496714	0.248357	−	0.968669i	$-0.420109\pi$
$337$	9.11846	0.496714	0.248357	+	0.968669i	$0.420109\pi$
$338$	34.5971	1.88183
$339$	0	0
$340$	−14.8380	−0.804701
$341$	−23.8522	−1.29167
$342$	0	0
$343$	1.00000	0.0539949
$344$	−4.38964	−0.236674
$345$	0	0
$346$	−9.80569	−0.527157
$347$	16.6694	0.894859	0.447429	−	0.894319i	$-0.352340\pi$
$347$	16.6694	0.894859	0.447429	+	0.894319i	$0.352340\pi$
$348$	0	0
$349$	30.0303	1.60749	0.803744	−	0.594976i	$-0.202838\pi$
$349$	30.0303	1.60749	0.803744	+	0.594976i	$0.202838\pi$
$350$	−0.0636955	−0.00340467
$351$	0	0
$352$	27.6440	1.47343
$353$	24.9661	1.32881	0.664405	−	0.747373i	$-0.268685\pi$
$353$	24.9661	1.32881	0.664405	+	0.747373i	$0.268685\pi$
$354$	0	0
$355$	−16.1206	−0.855594
$356$	25.8260	1.36877
$357$	0	0
$358$	4.43274	0.234278
$359$	15.4759	0.816789	0.408394	−	0.912806i	$-0.366089\pi$
$359$	15.4759	0.816789	0.408394	+	0.912806i	$0.366089\pi$
$360$	0	0
$361$	6.66098	0.350578
$362$	−23.3966	−1.22970
$363$	0	0
$364$	−13.5839	−0.711990
$365$	−17.6036	−0.921417
$366$	0	0
$367$	−6.79916	−0.354913	−0.177457	−	0.984129i	$-0.556787\pi$
$367$	−6.79916	−0.354913	−0.177457	+	0.984129i	$0.556787\pi$
$368$	−9.29053	−0.484302
$369$	0	0
$370$	−34.1242	−1.77403
$371$	0.167103	0.00867556
$372$	0	0
$373$	−9.46431	−0.490043	−0.245022	−	0.969518i	$-0.578795\pi$
$373$	−9.46431	−0.490043	−0.245022	+	0.969518i	$0.578795\pi$
$374$	19.6006	1.01352
$375$	0	0
$376$	−3.95018	−0.203715
$377$	−3.87171	−0.199403
$378$	0	0
$379$	3.61240	0.185556	0.0927782	−	0.995687i	$-0.470425\pi$
$379$	3.61240	0.185556	0.0927782	+	0.995687i	$0.470425\pi$
$380$	−28.3446	−1.45405
$381$	0	0
$382$	−2.42115	−0.123877
$383$	−5.60004	−0.286149	−0.143074	−	0.989712i	$-0.545699\pi$
$383$	−5.60004	−0.286149	−0.143074	+	0.989712i	$0.545699\pi$
$384$	0	0
$385$	−7.75931	−0.395451
$386$	−36.5501	−1.86035
$387$	0	0
$388$	−35.6135	−1.80800
$389$	−26.7575	−1.35666	−0.678329	−	0.734758i	$-0.737296\pi$
$389$	−26.7575	−1.35666	−0.678329	+	0.734758i	$0.737296\pi$
$390$	0	0
$391$	−9.05684	−0.458024
$392$	1.08284	0.0546916
$393$	0	0
$394$	6.80409	0.342785
$395$	−30.5869	−1.53899
$396$	0	0
$397$	32.4666	1.62945	0.814727	−	0.579844i	$-0.196887\pi$
$397$	32.4666	1.62945	0.814727	+	0.579844i	$0.196887\pi$
$398$	−29.4983	−1.47862
$399$	0	0
$400$	0.0815886	0.00407943
$401$	20.3956	1.01851	0.509253	−	0.860617i	$-0.329922\pi$
$401$	20.3956	1.01851	0.509253	+	0.860617i	$0.329922\pi$
$402$	0	0
$403$	−37.0897	−1.84757
$404$	−27.9348	−1.38981
$405$	0	0
$406$	1.51920	0.0753968
$407$	25.0868	1.24351
$408$	0	0
$409$	−24.5278	−1.21282	−0.606410	−	0.795152i	$-0.707391\pi$
$409$	−24.5278	−1.21282	−0.606410	+	0.795152i	$0.707391\pi$
$410$	−18.8608	−0.931470
$411$	0	0
$412$	43.9695	2.16622
$413$	12.1971	0.600178
$414$	0	0
$415$	−7.95913	−0.390698
$416$	42.9858	2.10755
$417$	0	0
$418$	37.4425	1.83137
$419$	0.366357	0.0178977	0.00894886	−	0.999960i	$-0.497151\pi$
$419$	0.366357	0.0178977	0.00894886	+	0.999960i	$0.497151\pi$
$420$	0	0
$421$	1.83510	0.0894372	0.0447186	−	0.999000i	$-0.485761\pi$
$421$	1.83510	0.0894372	0.0447186	+	0.999000i	$0.485761\pi$
$422$	29.7445	1.44794
$423$	0	0
$424$	0.180946	0.00878749
$425$	0.0795364	0.00385808
$426$	0	0
$427$	−3.87632	−0.187588
$428$	−18.0577	−0.872852
$429$	0	0
$430$	−19.1923	−0.925535
$431$	−35.6510	−1.71725	−0.858624	−	0.512606i	$-0.828680\pi$
$431$	−35.6510	−1.71725	−0.858624	+	0.512606i	$0.828680\pi$
$432$	0	0
$433$	−4.78359	−0.229885	−0.114942	−	0.993372i	$-0.536668\pi$
$433$	−4.78359	−0.229885	−0.114942	+	0.993372i	$0.536668\pi$
$434$	14.5535	0.698589
$435$	0	0
$436$	4.56965	0.218847
$437$	−17.3011	−0.827623
$438$	0	0
$439$	−20.3786	−0.972616	−0.486308	−	0.873788i	$-0.661657\pi$
$439$	−20.3786	−0.972616	−0.486308	+	0.873788i	$0.661657\pi$
$440$	−8.40208	−0.400554
$441$	0	0
$442$	30.4784	1.44971
$443$	−28.9756	−1.37667	−0.688336	−	0.725392i	$-0.741659\pi$
$443$	−28.9756	−1.37667	−0.688336	+	0.725392i	$0.741659\pi$
$444$	0	0
$445$	22.9393	1.08743
$446$	−12.3058	−0.582697
$447$	0	0
$448$	−11.4266	−0.539856
$449$	−12.8861	−0.608131	−0.304066	−	0.952651i	$-0.598344\pi$
$449$	−12.8861	−0.608131	−0.304066	+	0.952651i	$0.598344\pi$
$450$	0	0
$451$	13.8658	0.652913
$452$	−18.6920	−0.879200
$453$	0	0
$454$	0.636492	0.0298720
$455$	−12.0656	−0.565643
$456$	0	0
$457$	−6.94588	−0.324914	−0.162457	−	0.986716i	$-0.551942\pi$
$457$	−6.94588	−0.324914	−0.162457	+	0.986716i	$0.551942\pi$
$458$	52.5667	2.45628
$459$	0	0
$460$	19.1104	0.891028
$461$	38.2513	1.78154	0.890770	−	0.454454i	$-0.150166\pi$
$461$	38.2513	1.78154	0.890770	+	0.454454i	$0.150166\pi$
$462$	0	0
$463$	22.4537	1.04351	0.521755	−	0.853095i	$-0.325277\pi$
$463$	22.4537	1.04351	0.521755	+	0.853095i	$0.325277\pi$
$464$	−1.94597	−0.0903395
$465$	0	0
$466$	−13.8529	−0.641722
$467$	9.91332	0.458734	0.229367	−	0.973340i	$-0.426334\pi$
$467$	9.91332	0.458734	0.229367	+	0.973340i	$0.426334\pi$
$468$	0	0
$469$	14.7726	0.682134
$470$	−17.2709	−0.796648
$471$	0	0
$472$	13.2074	0.607921
$473$	14.1095	0.648754
$474$	0	0
$475$	0.151937	0.00697133
$476$	−6.65573	−0.305065
$477$	0	0
$478$	−23.5394	−1.07667
$479$	−24.9149	−1.13839	−0.569196	−	0.822202i	$-0.692745\pi$
$479$	−24.9149	−1.13839	−0.569196	+	0.822202i	$0.692745\pi$
$480$	0	0
$481$	39.0095	1.77868
$482$	12.5684	0.572477
$483$	0	0
$484$	2.79617	0.127099
$485$	−31.6328	−1.43637
$486$	0	0
$487$	4.54099	0.205772	0.102886	−	0.994693i	$-0.467192\pi$
$487$	4.54099	0.205772	0.102886	+	0.994693i	$0.467192\pi$
$488$	−4.19742	−0.190008
$489$	0	0
$490$	4.73436	0.213877
$491$	6.52601	0.294515	0.147257	−	0.989098i	$-0.452955\pi$
$491$	6.52601	0.294515	0.147257	+	0.989098i	$0.452955\pi$
$492$	0	0
$493$	−1.89703	−0.0854378
$494$	58.2223	2.61954
$495$	0	0
$496$	−18.6418	−0.837041
$497$	−7.23109	−0.324359
$498$	0	0
$499$	−1.55701	−0.0697015	−0.0348508	−	0.999393i	$-0.511096\pi$
$499$	−1.55701	−0.0697015	−0.0348508	+	0.999393i	$0.511096\pi$
$500$	−28.1450	−1.25868
$501$	0	0
$502$	−50.8308	−2.26869
$503$	0.292349	0.0130352	0.00651761	−	0.999979i	$-0.497925\pi$
$503$	0.292349	0.0130352	0.00651761	+	0.999979i	$0.497925\pi$
$504$	0	0
$505$	−24.8124	−1.10414
$506$	−25.2444	−1.12225
$507$	0	0
$508$	−2.50989	−0.111359
$509$	−16.3969	−0.726779	−0.363389	−	0.931637i	$-0.618381\pi$
$509$	−16.3969	−0.726779	−0.363389	+	0.931637i	$0.618381\pi$
$510$	0	0
$511$	−7.89631	−0.349312
$512$	27.4964	1.21518
$513$	0	0
$514$	43.8478	1.93404
$515$	39.0548	1.72096
$516$	0	0
$517$	12.6969	0.558410
$518$	−15.3068	−0.672542
$519$	0	0
$520$	−13.0651	−0.572941
$521$	−1.52643	−0.0668743	−0.0334371	−	0.999441i	$-0.510645\pi$
$521$	−1.52643	−0.0668743	−0.0334371	+	0.999441i	$0.510645\pi$
$522$	0	0
$523$	11.1354	0.486919	0.243460	−	0.969911i	$-0.421718\pi$
$523$	11.1354	0.486919	0.243460	+	0.969911i	$0.421718\pi$
$524$	−44.3891	−1.93915
$525$	0	0
$526$	−49.0804	−2.14001
$527$	−18.1729	−0.791623
$528$	0	0
$529$	−11.3353	−0.492840
$530$	0.791127	0.0343643
$531$	0	0
$532$	−12.7143	−0.551234
$533$	21.5610	0.933910
$534$	0	0
$535$	−16.0393	−0.693440
$536$	15.9963	0.690936
$537$	0	0
$538$	36.1676	1.55930
$539$	−3.48053	−0.149917
$540$	0	0
$541$	−25.6143	−1.10125	−0.550623	−	0.834754i	$-0.685610\pi$
$541$	−25.6143	−1.10125	−0.550623	+	0.834754i	$0.685610\pi$
$542$	54.8206	2.35475
$543$	0	0
$544$	21.0618	0.903018
$545$	4.05888	0.173863
$546$	0	0
$547$	−34.1531	−1.46028	−0.730140	−	0.683297i	$-0.760545\pi$
$547$	−34.1531	−1.46028	−0.730140	+	0.683297i	$0.760545\pi$
$548$	35.1032	1.49953
$549$	0	0
$550$	0.221694	0.00945306
$551$	−3.62384	−0.154381
$552$	0	0
$553$	−13.7201	−0.583437
$554$	43.0178	1.82765
$555$	0	0
$556$	35.3911	1.50092
$557$	−13.1463	−0.557028	−0.278514	−	0.960432i	$-0.589842\pi$
$557$	−13.1463	−0.557028	−0.278514	+	0.960432i	$0.589842\pi$
$558$	0	0
$559$	21.9399	0.927960
$560$	−6.06432	−0.256264
$561$	0	0
$562$	25.5580	1.07810
$563$	30.6198	1.29047	0.645234	−	0.763985i	$-0.276760\pi$
$563$	30.6198	1.29047	0.645234	+	0.763985i	$0.276760\pi$
$564$	0	0
$565$	−16.6027	−0.698482
$566$	20.9755	0.881666
$567$	0	0
$568$	−7.83010	−0.328544
$569$	11.1405	0.467035	0.233517	−	0.972353i	$-0.424976\pi$
$569$	11.1405	0.467035	0.233517	+	0.972353i	$0.424976\pi$
$570$	0	0
$571$	−22.9775	−0.961576	−0.480788	−	0.876837i	$-0.659649\pi$
$571$	−22.9775	−0.961576	−0.480788	+	0.876837i	$0.659649\pi$
$572$	47.2792	1.97684
$573$	0	0
$574$	−8.46023	−0.353123
$575$	−0.102438	−0.00427197
$576$	0	0
$577$	45.2387	1.88331	0.941655	−	0.336580i	$-0.109270\pi$
$577$	45.2387	1.88331	0.941655	+	0.336580i	$0.109270\pi$
$578$	−21.1685	−0.880493
$579$	0	0
$580$	4.00283	0.166208
$581$	−3.57015	−0.148115
$582$	0	0
$583$	−0.581606	−0.0240877
$584$	−8.55042	−0.353819
$585$	0	0
$586$	−58.0650	−2.39864
$587$	−7.60894	−0.314055	−0.157027	−	0.987594i	$-0.550191\pi$
$587$	−7.60894	−0.314055	−0.157027	+	0.987594i	$0.550191\pi$
$588$	0	0
$589$	−34.7152	−1.43042
$590$	57.7453	2.37734
$591$	0	0
$592$	19.6067	0.805831
$593$	−27.6048	−1.13359	−0.566796	−	0.823858i	$-0.691817\pi$
$593$	−27.6048	−1.13359	−0.566796	+	0.823858i	$0.691817\pi$
$594$	0	0
$595$	−5.91179	−0.242360
$596$	22.7335	0.931199
$597$	0	0
$598$	−39.2545	−1.60523
$599$	−14.1488	−0.578103	−0.289051	−	0.957314i	$-0.593340\pi$
$599$	−14.1488	−0.578103	−0.289051	+	0.957314i	$0.593340\pi$
$600$	0	0
$601$	−4.15288	−0.169400	−0.0846998	−	0.996407i	$-0.526993\pi$
$601$	−4.15288	−0.169400	−0.0846998	+	0.996407i	$0.526993\pi$
$602$	−8.60892	−0.350873
$603$	0	0
$604$	24.4263	0.993891
$605$	2.48363	0.100974
$606$	0	0
$607$	−21.1911	−0.860121	−0.430061	−	0.902800i	$-0.641508\pi$
$607$	−21.1911	−0.860121	−0.430061	+	0.902800i	$0.641508\pi$
$608$	40.2339	1.63170
$609$	0	0
$610$	−18.3519	−0.743047
$611$	19.7435	0.798735
$612$	0	0
$613$	−1.81134	−0.0731592	−0.0365796	−	0.999331i	$-0.511646\pi$
$613$	−1.81134	−0.0731592	−0.0365796	+	0.999331i	$0.511646\pi$
$614$	9.69202	0.391138
$615$	0	0
$616$	−3.76885	−0.151851
$617$	−40.8238	−1.64350	−0.821752	−	0.569845i	$-0.807003\pi$
$617$	−40.8238	−1.64350	−0.821752	+	0.569845i	$0.807003\pi$
$618$	0	0
$619$	−29.0154	−1.16623	−0.583114	−	0.812390i	$-0.698166\pi$
$619$	−29.0154	−1.16623	−0.583114	+	0.812390i	$0.698166\pi$
$620$	38.3458	1.54000
$621$	0	0
$622$	−50.5880	−2.02840
$623$	10.2897	0.412246
$624$	0	0
$625$	−24.8491	−0.993965
$626$	21.0168	0.840002
$627$	0	0
$628$	51.9102	2.07144
$629$	19.1136	0.762107
$630$	0	0
$631$	−15.9045	−0.633147	−0.316573	−	0.948568i	$-0.602532\pi$
$631$	−15.9045	−0.633147	−0.316573	+	0.948568i	$0.602532\pi$
$632$	−14.8566	−0.590965
$633$	0	0
$634$	−70.8933	−2.81553
$635$	−2.22935	−0.0884691
$636$	0	0
$637$	−5.41214	−0.214437
$638$	−5.28763	−0.209339
$639$	0	0
$640$	−18.6846	−0.738573
$641$	−46.6841	−1.84391	−0.921956	−	0.387294i	$-0.873410\pi$
$641$	−46.6841	−1.84391	−0.921956	+	0.387294i	$0.873410\pi$
$642$	0	0
$643$	28.2470	1.11395	0.556976	−	0.830529i	$-0.311962\pi$
$643$	28.2470	1.11395	0.556976	+	0.830529i	$0.311962\pi$
$644$	8.57219	0.337792
$645$	0	0
$646$	28.5273	1.12239
$647$	−23.2097	−0.912467	−0.456233	−	0.889860i	$-0.650802\pi$
$647$	−23.2097	−0.912467	−0.456233	+	0.889860i	$0.650802\pi$
$648$	0	0
$649$	−42.4522	−1.66639
$650$	0.344729	0.0135214
$651$	0	0
$652$	9.25775	0.362561
$653$	−6.45346	−0.252543	−0.126272	−	0.991996i	$-0.540301\pi$
$653$	−6.45346	−0.252543	−0.126272	+	0.991996i	$0.540301\pi$
$654$	0	0
$655$	−39.4275	−1.54056
$656$	10.8368	0.423108
$657$	0	0
$658$	−7.74706	−0.302012
$659$	−14.3202	−0.557836	−0.278918	−	0.960315i	$-0.589976\pi$
$659$	−14.3202	−0.557836	−0.278918	+	0.960315i	$0.589976\pi$
$660$	0	0
$661$	−43.0471	−1.67434	−0.837170	−	0.546943i	$-0.815791\pi$
$661$	−43.0471	−1.67434	−0.837170	+	0.546943i	$0.815791\pi$
$662$	−17.8668	−0.694412
$663$	0	0
$664$	−3.86590	−0.150026
$665$	−11.2931	−0.437930
$666$	0	0
$667$	2.44326	0.0946034
$668$	−6.96522	−0.269493
$669$	0	0
$670$	69.9388	2.70197
$671$	13.4916	0.520838
$672$	0	0
$673$	−10.3050	−0.397229	−0.198614	−	0.980078i	$-0.563644\pi$
$673$	−10.3050	−0.397229	−0.198614	+	0.980078i	$0.563644\pi$
$674$	19.3644	0.745890
$675$	0	0
$676$	40.8895	1.57267
$677$	27.3344	1.05054	0.525272	−	0.850934i	$-0.323963\pi$
$677$	27.3344	1.05054	0.525272	+	0.850934i	$0.323963\pi$
$678$	0	0
$679$	−14.1892	−0.544533
$680$	−6.40151	−0.245487
$681$	0	0
$682$	−50.6538	−1.93963
$683$	−24.4108	−0.934052	−0.467026	−	0.884244i	$-0.654675\pi$
$683$	−24.4108	−0.934052	−0.467026	+	0.884244i	$0.654675\pi$
$684$	0	0
$685$	31.1795	1.19131
$686$	2.12365	0.0810814
$687$	0	0
$688$	11.0273	0.420412
$689$	−0.904386	−0.0344544
$690$	0	0
$691$	−46.1899	−1.75715	−0.878573	−	0.477608i	$-0.841504\pi$
$691$	−46.1899	−1.75715	−0.878573	+	0.477608i	$0.841504\pi$
$692$	−11.5891	−0.440552
$693$	0	0
$694$	35.3999	1.34376
$695$	31.4352	1.19241
$696$	0	0
$697$	10.5643	0.400150
$698$	63.7740	2.41388
$699$	0	0
$700$	−0.0752803	−0.00284533
$701$	31.9239	1.20575	0.602874	−	0.797836i	$-0.294022\pi$
$701$	31.9239	1.20575	0.602874	+	0.797836i	$0.294022\pi$
$702$	0	0
$703$	36.5122	1.37708
$704$	39.7706	1.49891
$705$	0	0
$706$	53.0192	1.99540
$707$	−11.1299	−0.418582
$708$	0	0
$709$	9.53875	0.358235	0.179118	−	0.983828i	$-0.442676\pi$
$709$	9.53875	0.358235	0.179118	+	0.983828i	$0.442676\pi$
$710$	−34.2346	−1.28480
$711$	0	0
$712$	11.1420	0.417565
$713$	23.4056	0.876547
$714$	0	0
$715$	41.9945	1.57051
$716$	5.23895	0.195789
$717$	0	0
$718$	32.8655	1.22653
$719$	−26.2141	−0.977622	−0.488811	−	0.872390i	$-0.662569\pi$
$719$	−26.2141	−0.977622	−0.488811	+	0.872390i	$0.662569\pi$
$720$	0	0
$721$	17.5185	0.652421
$722$	14.1456	0.526444
$723$	0	0
$724$	−27.6519	−1.02768
$725$	−0.0214565	−0.000796874 0
$726$	0	0
$727$	−17.8525	−0.662112	−0.331056	−	0.943611i	$-0.607405\pi$
$727$	−17.8525	−0.662112	−0.331056	+	0.943611i	$0.607405\pi$
$728$	−5.86048	−0.217204
$729$	0	0
$730$	−37.3840	−1.38364
$731$	10.7499	0.397601
$732$	0	0
$733$	15.3021	0.565197	0.282599	−	0.959238i	$-0.408804\pi$
$733$	15.3021	0.565197	0.282599	+	0.959238i	$0.408804\pi$
$734$	−14.4391	−0.532955
$735$	0	0
$736$	−27.1264	−0.999892
$737$	−51.4164	−1.89395
$738$	0	0
$739$	−49.1744	−1.80891	−0.904456	−	0.426568i	$-0.859723\pi$
$739$	−49.1744	−1.80891	−0.904456	+	0.426568i	$0.859723\pi$
$740$	−40.3306	−1.48258
$741$	0	0
$742$	0.354869	0.0130276
$743$	13.2591	0.486430	0.243215	−	0.969972i	$-0.421798\pi$
$743$	13.2591	0.486430	0.243215	+	0.969972i	$0.421798\pi$
$744$	0	0
$745$	20.1924	0.739794
$746$	−20.0989	−0.735873
$747$	0	0
$748$	23.1654	0.847012
$749$	−7.19461	−0.262885
$750$	0	0
$751$	27.3861	0.999335	0.499667	−	0.866217i	$-0.333456\pi$
$751$	27.3861	0.999335	0.499667	+	0.866217i	$0.333456\pi$
$752$	9.92333	0.361867
$753$	0	0
$754$	−8.22215	−0.299433
$755$	21.6960	0.789599
$756$	0	0
$757$	22.5710	0.820357	0.410179	−	0.912005i	$-0.365466\pi$
$757$	22.5710	0.820357	0.410179	+	0.912005i	$0.365466\pi$
$758$	7.67147	0.278641
$759$	0	0
$760$	−12.2287	−0.443580
$761$	36.8017	1.33406	0.667031	−	0.745030i	$-0.267565\pi$
$761$	36.8017	1.33406	0.667031	+	0.745030i	$0.267565\pi$
$762$	0	0
$763$	1.82066	0.0659122
$764$	−2.86151	−0.103526
$765$	0	0
$766$	−11.8925	−0.429695
$767$	−66.0122	−2.38356
$768$	0	0
$769$	34.2552	1.23528	0.617638	−	0.786463i	$-0.288090\pi$
$769$	34.2552	1.23528	0.617638	+	0.786463i	$0.288090\pi$
$770$	−16.4781	−0.593829
$771$	0	0
$772$	−43.1977	−1.55472
$773$	35.3914	1.27294	0.636469	−	0.771302i	$-0.280394\pi$
$773$	35.3914	1.27294	0.636469	+	0.771302i	$0.280394\pi$
$774$	0	0
$775$	−0.205546	−0.00738344
$776$	−15.3646	−0.551558
$777$	0	0
$778$	−56.8236	−2.03722
$779$	20.1807	0.723048
$780$	0	0
$781$	25.1680	0.900581
$782$	−19.2336	−0.687791
$783$	0	0
$784$	−2.72022	−0.0971507
$785$	46.1079	1.64566
$786$	0	0
$787$	8.14845	0.290461	0.145230	−	0.989398i	$-0.453608\pi$
$787$	8.14845	0.290461	0.145230	+	0.989398i	$0.453608\pi$
$788$	8.04160	0.286470
$789$	0	0
$790$	−64.9559	−2.31103
$791$	−7.44734	−0.264797
$792$	0	0
$793$	20.9792	0.744993
$794$	68.9478	2.44687
$795$	0	0
$796$	−34.8634	−1.23570
$797$	11.0717	0.392179	0.196089	−	0.980586i	$-0.437176\pi$
$797$	11.0717	0.392179	0.196089	+	0.980586i	$0.437176\pi$
$798$	0	0
$799$	9.67373	0.342232
$800$	0.238222	0.00842241
$801$	0	0
$802$	43.3131	1.52944
$803$	27.4833	0.969865
$804$	0	0
$805$	7.61404	0.268360
$806$	−78.7655	−2.77440
$807$	0	0
$808$	−12.0519	−0.423983
$809$	13.7519	0.483489	0.241745	−	0.970340i	$-0.422280\pi$
$809$	13.7519	0.483489	0.241745	+	0.970340i	$0.422280\pi$
$810$	0	0
$811$	24.8521	0.872675	0.436337	−	0.899783i	$-0.356275\pi$
$811$	24.8521	0.872675	0.436337	+	0.899783i	$0.356275\pi$
$812$	1.79551	0.0630101
$813$	0	0
$814$	53.2757	1.86731
$815$	8.22296	0.288038
$816$	0	0
$817$	20.5354	0.718441
$818$	−52.0884	−1.82123
$819$	0	0
$820$	−22.2912	−0.778442
$821$	−29.3095	−1.02291	−0.511454	−	0.859311i	$-0.670893\pi$
$821$	−29.3095	−1.02291	−0.511454	+	0.859311i	$0.670893\pi$
$822$	0	0
$823$	37.2357	1.29796	0.648978	−	0.760808i	$-0.275197\pi$
$823$	37.2357	1.29796	0.648978	+	0.760808i	$0.275197\pi$
$824$	18.9696	0.660839
$825$	0	0
$826$	25.9023	0.901256
$827$	−26.8308	−0.932999	−0.466499	−	0.884522i	$-0.654485\pi$
$827$	−26.8308	−0.932999	−0.466499	+	0.884522i	$0.654485\pi$
$828$	0	0
$829$	48.6057	1.68815	0.844074	−	0.536227i	$-0.180151\pi$
$829$	48.6057	1.68815	0.844074	+	0.536227i	$0.180151\pi$
$830$	−16.9024	−0.586691
$831$	0	0
$832$	61.8424	2.14400
$833$	−2.65180	−0.0918793
$834$	0	0
$835$	−6.18669	−0.214099
$836$	44.2524	1.53050
$837$	0	0
$838$	0.778015	0.0268761
$839$	10.2550	0.354041	0.177021	−	0.984207i	$-0.443354\pi$
$839$	10.2550	0.354041	0.177021	+	0.984207i	$0.443354\pi$
$840$	0	0
$841$	−28.4882	−0.982353
$842$	3.89711	0.134303
$843$	0	0
$844$	35.1543	1.21006
$845$	36.3191	1.24941
$846$	0	0
$847$	1.11406	0.0382796
$848$	−0.454557	−0.0156095
$849$	0	0
$850$	0.168908	0.00579348
$851$	−24.6171	−0.843865
$852$	0	0
$853$	34.2400	1.17236	0.586178	−	0.810183i	$-0.300632\pi$
$853$	34.2400	1.17236	0.586178	+	0.810183i	$0.300632\pi$
$854$	−8.23195	−0.281691
$855$	0	0
$856$	−7.79060	−0.266277
$857$	−37.3928	−1.27731	−0.638657	−	0.769492i	$-0.720510\pi$
$857$	−37.3928	−1.27731	−0.638657	+	0.769492i	$0.720510\pi$
$858$	0	0
$859$	30.7959	1.05074	0.525371	−	0.850873i	$-0.323927\pi$
$859$	30.7959	1.05074	0.525371	+	0.850873i	$0.323927\pi$
$860$	−22.6829	−0.773482
$861$	0	0
$862$	−75.7103	−2.57870
$863$	33.3833	1.13638	0.568191	−	0.822897i	$-0.307644\pi$
$863$	33.3833	1.13638	0.568191	+	0.822897i	$0.307644\pi$
$864$	0	0
$865$	−10.2938	−0.349998
$866$	−10.1587	−0.345206
$867$	0	0
$868$	17.2004	0.583820
$869$	47.7531	1.61991
$870$	0	0
$871$	−79.9514	−2.70905
$872$	1.97148	0.0667626
$873$	0	0
$874$	−36.7414	−1.24280
$875$	−11.2136	−0.379090
$876$	0	0
$877$	13.7322	0.463702	0.231851	−	0.972751i	$-0.425522\pi$
$877$	13.7322	0.463702	0.231851	+	0.972751i	$0.425522\pi$
$878$	−43.2770	−1.46053
$879$	0	0
$880$	21.1070	0.711518
$881$	15.6596	0.527584	0.263792	−	0.964580i	$-0.415027\pi$
$881$	15.6596	0.527584	0.263792	+	0.964580i	$0.415027\pi$
$882$	0	0
$883$	−17.0634	−0.574229	−0.287115	−	0.957896i	$-0.592696\pi$
$883$	−17.0634	−0.574229	−0.287115	+	0.957896i	$0.592696\pi$
$884$	36.0218	1.21154
$885$	0	0
$886$	−61.5341	−2.06728
$887$	−37.7638	−1.26799	−0.633993	−	0.773339i	$-0.718585\pi$
$887$	−37.7638	−1.26799	−0.633993	+	0.773339i	$0.718585\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	48.7150	1.63293
$891$	0	0
$892$	−14.5440	−0.486968
$893$	18.4795	0.618393
$894$	0	0
$895$	4.65337	0.155545
$896$	−8.38117	−0.279995
$897$	0	0
$898$	−27.3655	−0.913200
$899$	4.90249	0.163507
$900$	0	0
$901$	−0.443123	−0.0147626
$902$	29.4461	0.980446
$903$	0	0
$904$	−8.06426	−0.268213
$905$	−24.5611	−0.816440
$906$	0	0
$907$	−27.6677	−0.918690	−0.459345	−	0.888258i	$-0.651916\pi$
$907$	−27.6677	−0.918690	−0.459345	+	0.888258i	$0.651916\pi$
$908$	0.752255	0.0249645
$909$	0	0
$910$	−25.6231	−0.849396
$911$	7.15429	0.237032	0.118516	−	0.992952i	$-0.462186\pi$
$911$	7.15429	0.237032	0.118516	+	0.992952i	$0.462186\pi$
$912$	0	0
$913$	12.4260	0.411241
$914$	−14.7506	−0.487907
$915$	0	0
$916$	62.1274	2.05275
$917$	−17.6856	−0.584032
$918$	0	0
$919$	−30.8988	−1.01926	−0.509629	−	0.860394i	$-0.670217\pi$
$919$	−30.8988	−1.01926	−0.509629	+	0.860394i	$0.670217\pi$
$920$	8.24477	0.271822
$921$	0	0
$922$	81.2324	2.67525
$923$	39.1357	1.28817
$924$	0	0
$925$	0.216186	0.00710814
$926$	47.6838	1.56699
$927$	0	0
$928$	−5.68184	−0.186515
$929$	30.1453	0.989035	0.494518	−	0.869168i	$-0.335345\pi$
$929$	30.1453	0.989035	0.494518	+	0.869168i	$0.335345\pi$
$930$	0	0
$931$	−5.06567	−0.166021
$932$	−16.3724	−0.536295
$933$	0	0
$934$	21.0524	0.688857
$935$	20.5761	0.672911
$936$	0	0
$937$	−28.3638	−0.926604	−0.463302	−	0.886201i	$-0.653335\pi$
$937$	−28.3638	−0.926604	−0.463302	+	0.886201i	$0.653335\pi$
$938$	31.3718	1.02433
$939$	0	0
$940$	−20.4121	−0.665769
$941$	17.5203	0.571147	0.285573	−	0.958357i	$-0.407816\pi$
$941$	17.5203	0.571147	0.285573	+	0.958357i	$0.407816\pi$
$942$	0	0
$943$	−13.6062	−0.443077
$944$	−33.1787	−1.07987
$945$	0	0
$946$	29.9636	0.974200
$947$	−21.0284	−0.683330	−0.341665	−	0.939822i	$-0.610991\pi$
$947$	−21.0284	−0.683330	−0.341665	+	0.939822i	$0.610991\pi$
$948$	0	0
$949$	42.7360	1.38727
$950$	0.322660	0.0104685
$951$	0	0
$952$	−2.87147	−0.0930648
$953$	−42.8972	−1.38958	−0.694789	−	0.719214i	$-0.744502\pi$
$953$	−42.8972	−1.38958	−0.694789	+	0.719214i	$0.744502\pi$
$954$	0	0
$955$	−2.54166	−0.0822462
$956$	−27.8207	−0.899786
$957$	0	0
$958$	−52.9106	−1.70946
$959$	13.9859	0.451629
$960$	0	0
$961$	15.9643	0.514976
$962$	82.8426	2.67095
$963$	0	0
$964$	14.8543	0.478426
$965$	−38.3693	−1.23515
$966$	0	0
$967$	23.4309	0.753488	0.376744	−	0.926317i	$-0.377044\pi$
$967$	23.4309	0.753488	0.376744	+	0.926317i	$0.377044\pi$
$968$	1.20635	0.0387735
$969$	0	0
$970$	−67.1770	−2.15692
$971$	5.53690	0.177688	0.0888438	−	0.996046i	$-0.471683\pi$
$971$	5.53690	0.177688	0.0888438	+	0.996046i	$0.471683\pi$
$972$	0	0
$973$	14.1006	0.452045
$974$	9.64347	0.308997
$975$	0	0
$976$	10.5444	0.337519
$977$	−28.8246	−0.922180	−0.461090	−	0.887353i	$-0.652541\pi$
$977$	−28.8246	−0.922180	−0.461090	+	0.887353i	$0.652541\pi$
$978$	0	0
$979$	−35.8134	−1.14460
$980$	5.59544	0.178740
$981$	0	0
$982$	13.8590	0.442258
$983$	29.9716	0.955945	0.477972	−	0.878375i	$-0.341372\pi$
$983$	29.9716	0.955945	0.477972	+	0.878375i	$0.341372\pi$
$984$	0	0
$985$	7.14275	0.227587
$986$	−4.02862	−0.128297
$987$	0	0
$988$	68.8116	2.18919
$989$	−13.8453	−0.440255
$990$	0	0
$991$	−31.0916	−0.987656	−0.493828	−	0.869560i	$-0.664403\pi$
$991$	−31.0916	−0.987656	−0.493828	+	0.869560i	$0.664403\pi$
$992$	−54.4301	−1.72816
$993$	0	0
$994$	−15.3563	−0.487072
$995$	−30.9665	−0.981705
$996$	0	0
$997$	−34.9784	−1.10778	−0.553888	−	0.832591i	$-0.686856\pi$
$997$	−34.9784	−1.10778	−0.553888	+	0.832591i	$0.686856\pi$
$998$	−3.30656	−0.104667
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.19		20
3.2	odd	2		889.2.a.d.1.2	✓	20
21.20	even	2		6223.2.a.l.1.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.2	✓	20	3.2	odd	2
6223.2.a.l.1.2		20	21.20	even	2
8001.2.a.w.1.19		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)

\(f(q)\)	\(=\)	\(q+2.12365 q^{2} +2.50989 q^{4} +2.22935 q^{5} +1.00000 q^{7} +1.08284 q^{8} +O(q^{10})\)	\(q+2.12365 q^{2} +2.50989 q^{4} +2.22935 q^{5} +1.00000 q^{7} +1.08284 q^{8} +4.73436 q^{10} -3.48053 q^{11} -5.41214 q^{13} +2.12365 q^{14} -2.72022 q^{16} -2.65180 q^{17} -5.06567 q^{19} +5.59544 q^{20} -7.39142 q^{22} +3.41536 q^{23} -0.0299934 q^{25} -11.4935 q^{26} +2.50989 q^{28} +0.715374 q^{29} +6.85305 q^{31} -7.94247 q^{32} -5.63149 q^{34} +2.22935 q^{35} -7.20777 q^{37} -10.7577 q^{38} +2.41403 q^{40} -3.98381 q^{41} -4.05383 q^{43} -8.73575 q^{44} +7.25303 q^{46} -3.64799 q^{47} +1.00000 q^{49} -0.0636955 q^{50} -13.5839 q^{52} +0.167103 q^{53} -7.75931 q^{55} +1.08284 q^{56} +1.51920 q^{58} +12.1971 q^{59} -3.87632 q^{61} +14.5535 q^{62} -11.4266 q^{64} -12.0656 q^{65} +14.7726 q^{67} -6.65573 q^{68} +4.73436 q^{70} -7.23109 q^{71} -7.89631 q^{73} -15.3068 q^{74} -12.7143 q^{76} -3.48053 q^{77} -13.7201 q^{79} -6.06432 q^{80} -8.46023 q^{82} -3.57015 q^{83} -5.91179 q^{85} -8.60892 q^{86} -3.76885 q^{88} +10.2897 q^{89} -5.41214 q^{91} +8.57219 q^{92} -7.74706 q^{94} -11.2931 q^{95} -14.1892 q^{97} +2.12365 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) 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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