LMFDB - Embedded newform 1040.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(1,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1040.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:	$8.30444181021$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	1040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.67282 q^{3} +1.00000 q^{5} +1.14399 q^{7} +4.14399 q^{9} +O(q^{10})$	$q+2.67282 q^{3} +1.00000 q^{5} +1.14399 q^{7} +4.14399 q^{9} -3.81681 q^{11} +1.00000 q^{13} +2.67282 q^{15} +7.34565 q^{17} -5.52884 q^{19} +3.05767 q^{21} +6.67282 q^{23} +1.00000 q^{25} +3.05767 q^{27} +2.85601 q^{29} +0.183190 q^{31} -10.2017 q^{33} +1.14399 q^{35} +6.48963 q^{37} +2.67282 q^{39} -7.34565 q^{41} -2.67282 q^{43} +4.14399 q^{45} +2.85601 q^{47} -5.69129 q^{49} +19.6336 q^{51} -13.6336 q^{53} -3.81681 q^{55} -14.7776 q^{57} -5.16246 q^{59} +5.14399 q^{61} +4.74066 q^{63} +1.00000 q^{65} +2.48963 q^{67} +17.8353 q^{69} -3.81681 q^{71} +12.7776 q^{73} +2.67282 q^{75} -4.36638 q^{77} -9.34565 q^{79} -4.25934 q^{81} -11.8353 q^{83} +7.34565 q^{85} +7.63362 q^{87} -14.9793 q^{89} +1.14399 q^{91} +0.489634 q^{93} -5.52884 q^{95} +7.71203 q^{97} -15.8168 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 3 q^{5} + 2 q^{7} + 11 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 + 3 * q^5 + 2 * q^7 + 11 * q^9	$ 3 q - 2 q^{3} + 3 q^{5} + 2 q^{7} + 11 q^{9} + 3 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} - 8 q^{21} + 10 q^{23} + 3 q^{25} - 8 q^{27} + 10 q^{29} + 12 q^{31} - 12 q^{33} + 2 q^{35} - 2 q^{37} - 2 q^{39} - 2 q^{41} + 2 q^{43} + 11 q^{45} + 10 q^{47} + 23 q^{49} + 36 q^{51} - 18 q^{53} - 20 q^{57} + 16 q^{59} + 14 q^{61} + 50 q^{63} + 3 q^{65} - 14 q^{67} + 12 q^{69} + 14 q^{73} - 2 q^{75} - 36 q^{77} - 8 q^{79} + 23 q^{81} + 6 q^{83} + 2 q^{85} - 2 q^{89} + 2 q^{91} - 20 q^{93} - 8 q^{95} + 26 q^{97} - 36 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 3 * q^5 + 2 * q^7 + 11 * q^9 + 3 * q^13 - 2 * q^15 + 2 * q^17 - 8 * q^19 - 8 * q^21 + 10 * q^23 + 3 * q^25 - 8 * q^27 + 10 * q^29 + 12 * q^31 - 12 * q^33 + 2 * q^35 - 2 * q^37 - 2 * q^39 - 2 * q^41 + 2 * q^43 + 11 * q^45 + 10 * q^47 + 23 * q^49 + 36 * q^51 - 18 * q^53 - 20 * q^57 + 16 * q^59 + 14 * q^61 + 50 * q^63 + 3 * q^65 - 14 * q^67 + 12 * q^69 + 14 * q^73 - 2 * q^75 - 36 * q^77 - 8 * q^79 + 23 * q^81 + 6 * q^83 + 2 * q^85 - 2 * q^89 + 2 * q^91 - 20 * q^93 - 8 * q^95 + 26 * q^97 - 36 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	2.67282	1.54316	0.771578	−	0.636135i	$-0.219468\pi$
$3$	2.67282	1.54316	0.771578	+	0.636135i	$0.219468\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.14399	0.432386	0.216193	−	0.976351i	$-0.430636\pi$
$7$	1.14399	0.432386	0.216193	+	0.976351i	$0.430636\pi$
$8$	0	0
$9$	4.14399	1.38133
$10$	0	0
$11$	−3.81681	−1.15081	−0.575406	−	0.817868i	$-0.695156\pi$
$11$	−3.81681	−1.15081	−0.575406	+	0.817868i	$0.695156\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.67282	0.690120
$16$	0	0
$17$	7.34565	1.78158	0.890791	−	0.454414i	$-0.150151\pi$
$17$	7.34565	1.78158	0.890791	+	0.454414i	$0.150151\pi$
$18$	0	0
$19$	−5.52884	−1.26840	−0.634201	−	0.773168i	$-0.718671\pi$
$19$	−5.52884	−1.26840	−0.634201	+	0.773168i	$0.718671\pi$
$20$	0	0
$21$	3.05767	0.667239
$22$	0	0
$23$	6.67282	1.39138	0.695690	−	0.718342i	$-0.255099\pi$
$23$	6.67282	1.39138	0.695690	+	0.718342i	$0.255099\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.05767	0.588450
$28$	0	0
$29$	2.85601	0.530348	0.265174	−	0.964201i	$-0.414571\pi$
$29$	2.85601	0.530348	0.265174	+	0.964201i	$0.414571\pi$
$30$	0	0
$31$	0.183190	0.0329019	0.0164509	−	0.999865i	$-0.494763\pi$
$31$	0.183190	0.0329019	0.0164509	+	0.999865i	$0.494763\pi$
$32$	0	0
$33$	−10.2017	−1.77588
$34$	0	0
$35$	1.14399	0.193369
$36$	0	0
$37$	6.48963	1.06689	0.533445	−	0.845835i	$-0.320897\pi$
$37$	6.48963	1.06689	0.533445	+	0.845835i	$0.320897\pi$
$38$	0	0
$39$	2.67282	0.427994
$40$	0	0
$41$	−7.34565	−1.14720	−0.573599	−	0.819136i	$-0.694453\pi$
$41$	−7.34565	−1.14720	−0.573599	+	0.819136i	$0.694453\pi$
$42$	0	0
$43$	−2.67282	−0.407602	−0.203801	−	0.979012i	$-0.565330\pi$
$43$	−2.67282	−0.407602	−0.203801	+	0.979012i	$0.565330\pi$
$44$	0	0
$45$	4.14399	0.617749
$46$	0	0
$47$	2.85601	0.416592	0.208296	−	0.978066i	$-0.433208\pi$
$47$	2.85601	0.416592	0.208296	+	0.978066i	$0.433208\pi$
$48$	0	0
$49$	−5.69129	−0.813042
$50$	0	0
$51$	19.6336	2.74926
$52$	0	0
$53$	−13.6336	−1.87272	−0.936361	−	0.351039i	$-0.885829\pi$
$53$	−13.6336	−1.87272	−0.936361	+	0.351039i	$0.885829\pi$
$54$	0	0
$55$	−3.81681	−0.514659
$56$	0	0
$57$	−14.7776	−1.95734
$58$	0	0
$59$	−5.16246	−0.672095	−0.336047	−	0.941845i	$-0.609090\pi$
$59$	−5.16246	−0.672095	−0.336047	+	0.941845i	$0.609090\pi$
$60$	0	0
$61$	5.14399	0.658620	0.329310	−	0.944222i	$-0.393184\pi$
$61$	5.14399	0.658620	0.329310	+	0.944222i	$0.393184\pi$
$62$	0	0
$63$	4.74066	0.597268
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	2.48963	0.304157	0.152079	−	0.988368i	$-0.451403\pi$
$67$	2.48963	0.304157	0.152079	+	0.988368i	$0.451403\pi$
$68$	0	0
$69$	17.8353	2.14712
$70$	0	0
$71$	−3.81681	−0.452972	−0.226486	−	0.974014i	$-0.572724\pi$
$71$	−3.81681	−0.452972	−0.226486	+	0.974014i	$0.572724\pi$
$72$	0	0
$73$	12.7776	1.49551	0.747753	−	0.663977i	$-0.231133\pi$
$73$	12.7776	1.49551	0.747753	+	0.663977i	$0.231133\pi$
$74$	0	0
$75$	2.67282	0.308631
$76$	0	0
$77$	−4.36638	−0.497595
$78$	0	0
$79$	−9.34565	−1.05147	−0.525734	−	0.850649i	$-0.676209\pi$
$79$	−9.34565	−1.05147	−0.525734	+	0.850649i	$0.676209\pi$
$80$	0	0
$81$	−4.25934	−0.473259
$82$	0	0
$83$	−11.8353	−1.29909	−0.649545	−	0.760323i	$-0.725041\pi$
$83$	−11.8353	−1.29909	−0.649545	+	0.760323i	$0.725041\pi$
$84$	0	0
$85$	7.34565	0.796747
$86$	0	0
$87$	7.63362	0.818410
$88$	0	0
$89$	−14.9793	−1.58780	−0.793900	−	0.608049i	$-0.791952\pi$
$89$	−14.9793	−1.58780	−0.793900	+	0.608049i	$0.791952\pi$
$90$	0	0
$91$	1.14399	0.119922
$92$	0	0
$93$	0.489634	0.0507727
$94$	0	0
$95$	−5.52884	−0.567247
$96$	0	0
$97$	7.71203	0.783038	0.391519	−	0.920170i	$-0.371950\pi$
$97$	7.71203	0.783038	0.391519	+	0.920170i	$0.371950\pi$
$98$	0	0
$99$	−15.8168	−1.58965
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	4.38485	0.432052	0.216026	−	0.976388i	$-0.430690\pi$
$103$	4.38485	0.432052	0.216026	+	0.976388i	$0.430690\pi$
$104$	0	0
$105$	3.05767	0.298398
$106$	0	0
$107$	−0.384851	−0.0372049	−0.0186025	−	0.999827i	$-0.505922\pi$
$107$	−0.384851	−0.0372049	−0.0186025	+	0.999827i	$0.505922\pi$
$108$	0	0
$109$	−12.6913	−1.21561	−0.607803	−	0.794088i	$-0.707949\pi$
$109$	−12.6913	−1.21561	−0.607803	+	0.794088i	$0.707949\pi$
$110$	0	0
$111$	17.3456	1.64638
$112$	0	0
$113$	1.05767	0.0994976	0.0497488	−	0.998762i	$-0.484158\pi$
$113$	1.05767	0.0994976	0.0497488	+	0.998762i	$0.484158\pi$
$114$	0	0
$115$	6.67282	0.622244
$116$	0	0
$117$	4.14399	0.383112
$118$	0	0
$119$	8.40332	0.770331
$120$	0	0
$121$	3.56804	0.324367
$122$	0	0
$123$	−19.6336	−1.77030
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.96080	−0.795142	−0.397571	−	0.917571i	$-0.630147\pi$
$127$	−8.96080	−0.795142	−0.397571	+	0.917571i	$0.630147\pi$
$128$	0	0
$129$	−7.14399	−0.628993
$130$	0	0
$131$	8.97927	0.784522	0.392261	−	0.919854i	$-0.371693\pi$
$131$	8.97927	0.784522	0.392261	+	0.919854i	$0.371693\pi$
$132$	0	0
$133$	−6.32492	−0.548440
$134$	0	0
$135$	3.05767	0.263163
$136$	0	0
$137$	−14.4033	−1.23056	−0.615279	−	0.788309i	$-0.710957\pi$
$137$	−14.4033	−1.23056	−0.615279	+	0.788309i	$0.710957\pi$
$138$	0	0
$139$	−21.3456	−1.81051	−0.905257	−	0.424864i	$-0.860322\pi$
$139$	−21.3456	−1.81051	−0.905257	+	0.424864i	$0.860322\pi$
$140$	0	0
$141$	7.63362	0.642867
$142$	0	0
$143$	−3.81681	−0.319178
$144$	0	0
$145$	2.85601	0.237179
$146$	0	0
$147$	−15.2118	−1.25465
$148$	0	0
$149$	−14.4033	−1.17997	−0.589983	−	0.807416i	$-0.700866\pi$
$149$	−14.4033	−1.17997	−0.589983	+	0.807416i	$0.700866\pi$
$150$	0	0
$151$	22.5081	1.83168	0.915842	−	0.401539i	$-0.131525\pi$
$151$	22.5081	1.83168	0.915842	+	0.401539i	$0.131525\pi$
$152$	0	0
$153$	30.4403	2.46095
$154$	0	0
$155$	0.183190	0.0147142
$156$	0	0
$157$	16.6913	1.33211	0.666055	−	0.745902i	$-0.267982\pi$
$157$	16.6913	1.33211	0.666055	+	0.745902i	$0.267982\pi$
$158$	0	0
$159$	−36.4403	−2.88990
$160$	0	0
$161$	7.63362	0.601614
$162$	0	0
$163$	23.4689	1.83823	0.919113	−	0.393994i	$-0.128907\pi$
$163$	23.4689	1.83823	0.919113	+	0.393994i	$0.128907\pi$
$164$	0	0
$165$	−10.2017	−0.794198
$166$	0	0
$167$	9.14399	0.707583	0.353791	−	0.935324i	$-0.384892\pi$
$167$	9.14399	0.707583	0.353791	+	0.935324i	$0.384892\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−22.9114	−1.75208
$172$	0	0
$173$	−10.0369	−0.763095	−0.381547	−	0.924349i	$-0.624609\pi$
$173$	−10.0369	−0.763095	−0.381547	+	0.924349i	$0.624609\pi$
$174$	0	0
$175$	1.14399	0.0864773
$176$	0	0
$177$	−13.7983	−1.03715
$178$	0	0
$179$	20.9793	1.56806	0.784032	−	0.620720i	$-0.213160\pi$
$179$	20.9793	1.56806	0.784032	+	0.620720i	$0.213160\pi$
$180$	0	0
$181$	13.5473	1.00696	0.503482	−	0.864006i	$-0.332052\pi$
$181$	13.5473	1.00696	0.503482	+	0.864006i	$0.332052\pi$
$182$	0	0
$183$	13.7490	1.01635
$184$	0	0
$185$	6.48963	0.477127
$186$	0	0
$187$	−28.0369	−2.05026
$188$	0	0
$189$	3.49794	0.254438
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−3.71203	−0.267198	−0.133599	−	0.991036i	$-0.542653\pi$
$193$	−3.71203	−0.267198	−0.133599	+	0.991036i	$0.542653\pi$
$194$	0	0
$195$	2.67282	0.191405
$196$	0	0
$197$	8.69129	0.619229	0.309615	−	0.950862i	$-0.399800\pi$
$197$	8.69129	0.619229	0.309615	+	0.950862i	$0.399800\pi$
$198$	0	0
$199$	24.9793	1.77073	0.885367	−	0.464893i	$-0.153907\pi$
$199$	24.9793	1.77073	0.885367	+	0.464893i	$0.153907\pi$
$200$	0	0
$201$	6.65435	0.469362
$202$	0	0
$203$	3.26724	0.229315
$204$	0	0
$205$	−7.34565	−0.513042
$206$	0	0
$207$	27.6521	1.92195
$208$	0	0
$209$	21.1025	1.45969
$210$	0	0
$211$	−16.9793	−1.16890	−0.584451	−	0.811429i	$-0.698690\pi$
$211$	−16.9793	−1.16890	−0.584451	+	0.811429i	$0.698690\pi$
$212$	0	0
$213$	−10.2017	−0.699006
$214$	0	0
$215$	−2.67282	−0.182285
$216$	0	0
$217$	0.209567	0.0142263
$218$	0	0
$219$	34.1523	2.30780
$220$	0	0
$221$	7.34565	0.494122
$222$	0	0
$223$	−13.5473	−0.907195	−0.453597	−	0.891207i	$-0.649860\pi$
$223$	−13.5473	−0.907195	−0.453597	+	0.891207i	$0.649860\pi$
$224$	0	0
$225$	4.14399	0.276266
$226$	0	0
$227$	−9.91369	−0.657995	−0.328997	−	0.944331i	$-0.606711\pi$
$227$	−9.91369	−0.657995	−0.328997	+	0.944331i	$0.606711\pi$
$228$	0	0
$229$	−14.0369	−0.927587	−0.463794	−	0.885943i	$-0.653512\pi$
$229$	−14.0369	−0.927587	−0.463794	+	0.885943i	$0.653512\pi$
$230$	0	0
$231$	−11.6706	−0.767867
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	2.85601	0.186306
$236$	0	0
$237$	−24.9793	−1.62258
$238$	0	0
$239$	10.1048	0.653624	0.326812	−	0.945089i	$-0.394026\pi$
$239$	10.1048	0.653624	0.326812	+	0.945089i	$0.394026\pi$
$240$	0	0
$241$	−26.0369	−1.67719	−0.838593	−	0.544758i	$-0.816622\pi$
$241$	−26.0369	−1.67719	−0.838593	+	0.544758i	$0.816622\pi$
$242$	0	0
$243$	−20.5575	−1.31876
$244$	0	0
$245$	−5.69129	−0.363603
$246$	0	0
$247$	−5.52884	−0.351791
$248$	0	0
$249$	−31.6336	−2.00470
$250$	0	0
$251$	−4.94233	−0.311957	−0.155978	−	0.987760i	$-0.549853\pi$
$251$	−4.94233	−0.311957	−0.155978	+	0.987760i	$0.549853\pi$
$252$	0	0
$253$	−25.4689	−1.60122
$254$	0	0
$255$	19.6336	1.22951
$256$	0	0
$257$	−12.2880	−0.766503	−0.383251	−	0.923644i	$-0.625196\pi$
$257$	−12.2880	−0.766503	−0.383251	+	0.923644i	$0.625196\pi$
$258$	0	0
$259$	7.42405	0.461308
$260$	0	0
$261$	11.8353	0.732586
$262$	0	0
$263$	7.44252	0.458926	0.229463	−	0.973317i	$-0.426303\pi$
$263$	7.44252	0.458926	0.229463	+	0.973317i	$0.426303\pi$
$264$	0	0
$265$	−13.6336	−0.837507
$266$	0	0
$267$	−40.0369	−2.45022
$268$	0	0
$269$	−21.2672	−1.29669	−0.648343	−	0.761348i	$-0.724538\pi$
$269$	−21.2672	−1.29669	−0.648343	+	0.761348i	$0.724538\pi$
$270$	0	0
$271$	19.8168	1.20379	0.601893	−	0.798577i	$-0.294413\pi$
$271$	19.8168	1.20379	0.601893	+	0.798577i	$0.294413\pi$
$272$	0	0
$273$	3.05767	0.185059
$274$	0	0
$275$	−3.81681	−0.230162
$276$	0	0
$277$	−11.9216	−0.716299	−0.358150	−	0.933664i	$-0.616592\pi$
$277$	−11.9216	−0.716299	−0.358150	+	0.933664i	$0.616592\pi$
$278$	0	0
$279$	0.759136	0.0454483
$280$	0	0
$281$	19.3456	1.15406	0.577032	−	0.816721i	$-0.304211\pi$
$281$	19.3456	1.15406	0.577032	+	0.816721i	$0.304211\pi$
$282$	0	0
$283$	−29.3641	−1.74552	−0.872758	−	0.488153i	$-0.837671\pi$
$283$	−29.3641	−1.74552	−0.872758	+	0.488153i	$0.837671\pi$
$284$	0	0
$285$	−14.7776	−0.875350
$286$	0	0
$287$	−8.40332	−0.496032
$288$	0	0
$289$	36.9585	2.17403
$290$	0	0
$291$	20.6129	1.20835
$292$	0	0
$293$	−24.6050	−1.43744	−0.718719	−	0.695300i	$-0.755271\pi$
$293$	−24.6050	−1.43744	−0.718719	+	0.695300i	$0.755271\pi$
$294$	0	0
$295$	−5.16246	−0.300570
$296$	0	0
$297$	−11.6706	−0.677195
$298$	0	0
$299$	6.67282	0.385899
$300$	0	0
$301$	−3.05767	−0.176241
$302$	0	0
$303$	16.0369	0.921298
$304$	0	0
$305$	5.14399	0.294544
$306$	0	0
$307$	15.2593	0.870896	0.435448	−	0.900214i	$-0.356590\pi$
$307$	15.2593	0.870896	0.435448	+	0.900214i	$0.356590\pi$
$308$	0	0
$309$	11.7199	0.666724
$310$	0	0
$311$	−16.0369	−0.909372	−0.454686	−	0.890652i	$-0.650248\pi$
$311$	−16.0369	−0.909372	−0.454686	+	0.890652i	$0.650248\pi$
$312$	0	0
$313$	27.9216	1.57822	0.789111	−	0.614251i	$-0.210542\pi$
$313$	27.9216	1.57822	0.789111	+	0.614251i	$0.210542\pi$
$314$	0	0
$315$	4.74066	0.267106
$316$	0	0
$317$	−21.9137	−1.23080	−0.615398	−	0.788217i	$-0.711005\pi$
$317$	−21.9137	−1.23080	−0.615398	+	0.788217i	$0.711005\pi$
$318$	0	0
$319$	−10.9009	−0.610331
$320$	0	0
$321$	−1.02864	−0.0574130
$322$	0	0
$323$	−40.6129	−2.25976
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−33.9216	−1.87587
$328$	0	0
$329$	3.26724	0.180129
$330$	0	0
$331$	7.24086	0.397994	0.198997	−	0.980000i	$-0.436232\pi$
$331$	7.24086	0.397994	0.198997	+	0.980000i	$0.436232\pi$
$332$	0	0
$333$	26.8930	1.47372
$334$	0	0
$335$	2.48963	0.136023
$336$	0	0
$337$	9.05767	0.493403	0.246701	−	0.969092i	$-0.420653\pi$
$337$	9.05767	0.493403	0.246701	+	0.969092i	$0.420653\pi$
$338$	0	0
$339$	2.82698	0.153540
$340$	0	0
$341$	−0.699201	−0.0378638
$342$	0	0
$343$	−14.5187	−0.783935
$344$	0	0
$345$	17.8353	0.960219
$346$	0	0
$347$	2.63588	0.141502	0.0707508	−	0.997494i	$-0.477461\pi$
$347$	2.63588	0.141502	0.0707508	+	0.997494i	$0.477461\pi$
$348$	0	0
$349$	−17.6336	−0.943906	−0.471953	−	0.881624i	$-0.656451\pi$
$349$	−17.6336	−0.943906	−0.471953	+	0.881624i	$0.656451\pi$
$350$	0	0
$351$	3.05767	0.163207
$352$	0	0
$353$	12.6050	0.670896	0.335448	−	0.942059i	$-0.391112\pi$
$353$	12.6050	0.670896	0.335448	+	0.942059i	$0.391112\pi$
$354$	0	0
$355$	−3.81681	−0.202575
$356$	0	0
$357$	22.4606	1.18874
$358$	0	0
$359$	−6.83754	−0.360872	−0.180436	−	0.983587i	$-0.557751\pi$
$359$	−6.83754	−0.360872	−0.180436	+	0.983587i	$0.557751\pi$
$360$	0	0
$361$	11.5680	0.608844
$362$	0	0
$363$	9.53674	0.500549
$364$	0	0
$365$	12.7776	0.668811
$366$	0	0
$367$	10.0969	0.527053	0.263526	−	0.964652i	$-0.415114\pi$
$367$	10.0969	0.527053	0.263526	+	0.964652i	$0.415114\pi$
$368$	0	0
$369$	−30.4403	−1.58466
$370$	0	0
$371$	−15.5967	−0.809739
$372$	0	0
$373$	1.42405	0.0737347	0.0368674	−	0.999320i	$-0.488262\pi$
$373$	1.42405	0.0737347	0.0368674	+	0.999320i	$0.488262\pi$
$374$	0	0
$375$	2.67282	0.138024
$376$	0	0
$377$	2.85601	0.147092
$378$	0	0
$379$	24.4297	1.25487	0.627435	−	0.778669i	$-0.284105\pi$
$379$	24.4297	1.25487	0.627435	+	0.778669i	$0.284105\pi$
$380$	0	0
$381$	−23.9506	−1.22703
$382$	0	0
$383$	−5.54731	−0.283454	−0.141727	−	0.989906i	$-0.545266\pi$
$383$	−5.54731	−0.283454	−0.141727	+	0.989906i	$0.545266\pi$
$384$	0	0
$385$	−4.36638	−0.222531
$386$	0	0
$387$	−11.0761	−0.563032
$388$	0	0
$389$	34.8066	1.76477	0.882383	−	0.470531i	$-0.155938\pi$
$389$	34.8066	1.76477	0.882383	+	0.470531i	$0.155938\pi$
$390$	0	0
$391$	49.0162	2.47886
$392$	0	0
$393$	24.0000	1.21064
$394$	0	0
$395$	−9.34565	−0.470231
$396$	0	0
$397$	15.2224	0.763990	0.381995	−	0.924164i	$-0.375237\pi$
$397$	15.2224	0.763990	0.381995	+	0.924164i	$0.375237\pi$
$398$	0	0
$399$	−16.9054	−0.846328
$400$	0	0
$401$	−0.863919	−0.0431421	−0.0215710	−	0.999767i	$-0.506867\pi$
$401$	−0.863919	−0.0431421	−0.0215710	+	0.999767i	$0.506867\pi$
$402$	0	0
$403$	0.183190	0.00912533
$404$	0	0
$405$	−4.25934	−0.211648
$406$	0	0
$407$	−24.7697	−1.22779
$408$	0	0
$409$	6.03694	0.298508	0.149254	−	0.988799i	$-0.452313\pi$
$409$	6.03694	0.298508	0.149254	+	0.988799i	$0.452313\pi$
$410$	0	0
$411$	−38.4975	−1.89894
$412$	0	0
$413$	−5.90578	−0.290605
$414$	0	0
$415$	−11.8353	−0.580971
$416$	0	0
$417$	−57.0532	−2.79390
$418$	0	0
$419$	4.36638	0.213312	0.106656	−	0.994296i	$-0.465986\pi$
$419$	4.36638	0.213312	0.106656	+	0.994296i	$0.465986\pi$
$420$	0	0
$421$	34.6498	1.68873	0.844365	−	0.535769i	$-0.179978\pi$
$421$	34.6498	1.68873	0.844365	+	0.535769i	$0.179978\pi$
$422$	0	0
$423$	11.8353	0.575451
$424$	0	0
$425$	7.34565	0.356316
$426$	0	0
$427$	5.88465	0.284778
$428$	0	0
$429$	−10.2017	−0.492541
$430$	0	0
$431$	−15.8168	−0.761869	−0.380934	−	0.924602i	$-0.624398\pi$
$431$	−15.8168	−0.761869	−0.380934	+	0.924602i	$0.624398\pi$
$432$	0	0
$433$	7.38259	0.354785	0.177392	−	0.984140i	$-0.443234\pi$
$433$	7.38259	0.354785	0.177392	+	0.984140i	$0.443234\pi$
$434$	0	0
$435$	7.63362	0.366004
$436$	0	0
$437$	−36.8930	−1.76483
$438$	0	0
$439$	21.3826	1.02054	0.510268	−	0.860016i	$-0.329546\pi$
$439$	21.3826	1.02054	0.510268	+	0.860016i	$0.329546\pi$
$440$	0	0
$441$	−23.5846	−1.12308
$442$	0	0
$443$	7.24877	0.344399	0.172200	−	0.985062i	$-0.444913\pi$
$443$	7.24877	0.344399	0.172200	+	0.985062i	$0.444913\pi$
$444$	0	0
$445$	−14.9793	−0.710085
$446$	0	0
$447$	−38.4975	−1.82087
$448$	0	0
$449$	−28.9009	−1.36392	−0.681958	−	0.731391i	$-0.738871\pi$
$449$	−28.9009	−1.36392	−0.681958	+	0.731391i	$0.738871\pi$
$450$	0	0
$451$	28.0369	1.32021
$452$	0	0
$453$	60.1602	2.82657
$454$	0	0
$455$	1.14399	0.0536309
$456$	0	0
$457$	22.9793	1.07492	0.537462	−	0.843288i	$-0.319383\pi$
$457$	22.9793	1.07492	0.537462	+	0.843288i	$0.319383\pi$
$458$	0	0
$459$	22.4606	1.04837
$460$	0	0
$461$	12.4817	0.581332	0.290666	−	0.956825i	$-0.406123\pi$
$461$	12.4817	0.581332	0.290666	+	0.956825i	$0.406123\pi$
$462$	0	0
$463$	32.2017	1.49654	0.748269	−	0.663395i	$-0.230885\pi$
$463$	32.2017	1.49654	0.748269	+	0.663395i	$0.230885\pi$
$464$	0	0
$465$	0.489634	0.0227062
$466$	0	0
$467$	27.0761	1.25293	0.626467	−	0.779448i	$-0.284500\pi$
$467$	27.0761	1.25293	0.626467	+	0.779448i	$0.284500\pi$
$468$	0	0
$469$	2.84811	0.131513
$470$	0	0
$471$	44.6129	2.05565
$472$	0	0
$473$	10.2017	0.469073
$474$	0	0
$475$	−5.52884	−0.253680
$476$	0	0
$477$	−56.4975	−2.58684
$478$	0	0
$479$	−24.7961	−1.13296	−0.566481	−	0.824075i	$-0.691695\pi$
$479$	−24.7961	−1.13296	−0.566481	+	0.824075i	$0.691695\pi$
$480$	0	0
$481$	6.48963	0.295902
$482$	0	0
$483$	20.4033	0.928383
$484$	0	0
$485$	7.71203	0.350185
$486$	0	0
$487$	7.62571	0.345554	0.172777	−	0.984961i	$-0.444726\pi$
$487$	7.62571	0.345554	0.172777	+	0.984961i	$0.444726\pi$
$488$	0	0
$489$	62.7282	2.83667
$490$	0	0
$491$	34.3249	1.54906	0.774531	−	0.632536i	$-0.217986\pi$
$491$	34.3249	1.54906	0.774531	+	0.632536i	$0.217986\pi$
$492$	0	0
$493$	20.9793	0.944859
$494$	0	0
$495$	−15.8168	−0.710913
$496$	0	0
$497$	−4.36638	−0.195859
$498$	0	0
$499$	8.58651	0.384385	0.192193	−	0.981357i	$-0.438440\pi$
$499$	8.58651	0.384385	0.192193	+	0.981357i	$0.438440\pi$
$500$	0	0
$501$	24.4403	1.09191
$502$	0	0
$503$	−40.2280	−1.79368	−0.896840	−	0.442356i	$-0.854143\pi$
$503$	−40.2280	−1.79368	−0.896840	+	0.442356i	$0.854143\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	2.67282	0.118704
$508$	0	0
$509$	−10.3664	−0.459482	−0.229741	−	0.973252i	$-0.573788\pi$
$509$	−10.3664	−0.459482	−0.229741	+	0.973252i	$0.573788\pi$
$510$	0	0
$511$	14.6174	0.646636
$512$	0	0
$513$	−16.9054	−0.746391
$514$	0	0
$515$	4.38485	0.193220
$516$	0	0
$517$	−10.9009	−0.479419
$518$	0	0
$519$	−26.8270	−1.17757
$520$	0	0
$521$	32.8145	1.43763	0.718816	−	0.695201i	$-0.244684\pi$
$521$	32.8145	1.43763	0.718816	+	0.695201i	$0.244684\pi$
$522$	0	0
$523$	25.9401	1.13428	0.567140	−	0.823621i	$-0.308050\pi$
$523$	25.9401	1.13428	0.567140	+	0.823621i	$0.308050\pi$
$524$	0	0
$525$	3.05767	0.133448
$526$	0	0
$527$	1.34565	0.0586173
$528$	0	0
$529$	21.5266	0.935938
$530$	0	0
$531$	−21.3932	−0.928384
$532$	0	0
$533$	−7.34565	−0.318175
$534$	0	0
$535$	−0.384851	−0.0166385
$536$	0	0
$537$	56.0739	2.41977
$538$	0	0
$539$	21.7226	0.935658
$540$	0	0
$541$	6.61289	0.284310	0.142155	−	0.989844i	$-0.454597\pi$
$541$	6.61289	0.284310	0.142155	+	0.989844i	$0.454597\pi$
$542$	0	0
$543$	36.2096	1.55390
$544$	0	0
$545$	−12.6913	−0.543635
$546$	0	0
$547$	−8.38485	−0.358510	−0.179255	−	0.983803i	$-0.557369\pi$
$547$	−8.38485	−0.358510	−0.179255	+	0.983803i	$0.557369\pi$
$548$	0	0
$549$	21.3166	0.909771
$550$	0	0
$551$	−15.7904	−0.672695
$552$	0	0
$553$	−10.6913	−0.454640
$554$	0	0
$555$	17.3456	0.736282
$556$	0	0
$557$	40.4482	1.71384	0.856922	−	0.515446i	$-0.172374\pi$
$557$	40.4482	1.71384	0.856922	+	0.515446i	$0.172374\pi$
$558$	0	0
$559$	−2.67282	−0.113048
$560$	0	0
$561$	−74.9378	−3.16388
$562$	0	0
$563$	16.7512	0.705980	0.352990	−	0.935627i	$-0.385165\pi$
$563$	16.7512	0.705980	0.352990	+	0.935627i	$0.385165\pi$
$564$	0	0
$565$	1.05767	0.0444967
$566$	0	0
$567$	−4.87262	−0.204631
$568$	0	0
$569$	−12.4112	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$569$	−12.4112	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$570$	0	0
$571$	−12.0369	−0.503730	−0.251865	−	0.967762i	$-0.581044\pi$
$571$	−12.0369	−0.503730	−0.251865	+	0.967762i	$0.581044\pi$
$572$	0	0
$573$	−32.0739	−1.33991
$574$	0	0
$575$	6.67282	0.278276
$576$	0	0
$577$	−16.6050	−0.691274	−0.345637	−	0.938368i	$-0.612337\pi$
$577$	−16.6050	−0.691274	−0.345637	+	0.938368i	$0.612337\pi$
$578$	0	0
$579$	−9.92159	−0.412327
$580$	0	0
$581$	−13.5394	−0.561709
$582$	0	0
$583$	52.0369	2.15515
$584$	0	0
$585$	4.14399	0.171333
$586$	0	0
$587$	24.8515	1.02573	0.512865	−	0.858469i	$-0.328584\pi$
$587$	24.8515	1.02573	0.512865	+	0.858469i	$0.328584\pi$
$588$	0	0
$589$	−1.01283	−0.0417328
$590$	0	0
$591$	23.2303	0.955567
$592$	0	0
$593$	−15.5552	−0.638776	−0.319388	−	0.947624i	$-0.603477\pi$
$593$	−15.5552	−0.638776	−0.319388	+	0.947624i	$0.603477\pi$
$594$	0	0
$595$	8.40332	0.344503
$596$	0	0
$597$	66.7652	2.73252
$598$	0	0
$599$	0.193755	0.00791662	0.00395831	−	0.999992i	$-0.498740\pi$
$599$	0.193755	0.00791662	0.00395831	+	0.999992i	$0.498740\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	10.3170	0.420141
$604$	0	0
$605$	3.56804	0.145061
$606$	0	0
$607$	16.9608	0.688418	0.344209	−	0.938893i	$-0.388147\pi$
$607$	16.9608	0.688418	0.344209	+	0.938893i	$0.388147\pi$
$608$	0	0
$609$	8.73276	0.353869
$610$	0	0
$611$	2.85601	0.115542
$612$	0	0
$613$	−26.8066	−1.08271	−0.541355	−	0.840794i	$-0.682089\pi$
$613$	−26.8066	−1.08271	−0.541355	+	0.840794i	$0.682089\pi$
$614$	0	0
$615$	−19.6336	−0.791704
$616$	0	0
$617$	−0.287973	−0.0115934	−0.00579668	−	0.999983i	$-0.501845\pi$
$617$	−0.287973	−0.0115934	−0.00579668	+	0.999983i	$0.501845\pi$
$618$	0	0
$619$	−12.0106	−0.482745	−0.241373	−	0.970432i	$-0.577598\pi$
$619$	−12.0106	−0.482745	−0.241373	+	0.970432i	$0.577598\pi$
$620$	0	0
$621$	20.4033	0.818757
$622$	0	0
$623$	−17.1361	−0.686543
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	56.4033	2.25253
$628$	0	0
$629$	47.6706	1.90075
$630$	0	0
$631$	20.3928	0.811823	0.405911	−	0.913912i	$-0.366954\pi$
$631$	20.3928	0.811823	0.405911	+	0.913912i	$0.366954\pi$
$632$	0	0
$633$	−45.3826	−1.80380
$634$	0	0
$635$	−8.96080	−0.355598
$636$	0	0
$637$	−5.69129	−0.225497
$638$	0	0
$639$	−15.8168	−0.625703
$640$	0	0
$641$	21.2672	0.840006	0.420003	−	0.907523i	$-0.362029\pi$
$641$	21.2672	0.840006	0.420003	+	0.907523i	$0.362029\pi$
$642$	0	0
$643$	23.6627	0.933164	0.466582	−	0.884478i	$-0.345485\pi$
$643$	23.6627	0.933164	0.466582	+	0.884478i	$0.345485\pi$
$644$	0	0
$645$	−7.14399	−0.281294
$646$	0	0
$647$	39.8458	1.56650	0.783251	−	0.621706i	$-0.213560\pi$
$647$	39.8458	1.56650	0.783251	+	0.621706i	$0.213560\pi$
$648$	0	0
$649$	19.7041	0.773454
$650$	0	0
$651$	0.560135	0.0219534
$652$	0	0
$653$	−11.7120	−0.458327	−0.229164	−	0.973388i	$-0.573599\pi$
$653$	−11.7120	−0.458327	−0.229164	+	0.973388i	$0.573599\pi$
$654$	0	0
$655$	8.97927	0.350849
$656$	0	0
$657$	52.9502	2.06579
$658$	0	0
$659$	−21.1730	−0.824784	−0.412392	−	0.911007i	$-0.635307\pi$
$659$	−21.1730	−0.824784	−0.412392	+	0.911007i	$0.635307\pi$
$660$	0	0
$661$	2.57595	0.100193	0.0500963	−	0.998744i	$-0.484047\pi$
$661$	2.57595	0.100193	0.0500963	+	0.998744i	$0.484047\pi$
$662$	0	0
$663$	19.6336	0.762507
$664$	0	0
$665$	−6.32492	−0.245270
$666$	0	0
$667$	19.0577	0.737916
$668$	0	0
$669$	−36.2096	−1.39994
$670$	0	0
$671$	−19.6336	−0.757948
$672$	0	0
$673$	−16.7282	−0.644826	−0.322413	−	0.946599i	$-0.604494\pi$
$673$	−16.7282	−0.644826	−0.322413	+	0.946599i	$0.604494\pi$
$674$	0	0
$675$	3.05767	0.117690
$676$	0	0
$677$	−10.9423	−0.420548	−0.210274	−	0.977643i	$-0.567436\pi$
$677$	−10.9423	−0.420548	−0.210274	+	0.977643i	$0.567436\pi$
$678$	0	0
$679$	8.82245	0.338575
$680$	0	0
$681$	−26.4975	−1.01539
$682$	0	0
$683$	−6.89296	−0.263752	−0.131876	−	0.991266i	$-0.542100\pi$
$683$	−6.89296	−0.263752	−0.131876	+	0.991266i	$0.542100\pi$
$684$	0	0
$685$	−14.4033	−0.550323
$686$	0	0
$687$	−37.5183	−1.43141
$688$	0	0
$689$	−13.6336	−0.519400
$690$	0	0
$691$	−43.4504	−1.65293	−0.826466	−	0.562986i	$-0.809652\pi$
$691$	−43.4504	−1.65293	−0.826466	+	0.562986i	$0.809652\pi$
$692$	0	0
$693$	−18.0942	−0.687343
$694$	0	0
$695$	−21.3456	−0.809687
$696$	0	0
$697$	−53.9585	−2.04383
$698$	0	0
$699$	−16.0369	−0.606573
$700$	0	0
$701$	32.1153	1.21298	0.606490	−	0.795091i	$-0.292577\pi$
$701$	32.1153	1.21298	0.606490	+	0.795091i	$0.292577\pi$
$702$	0	0
$703$	−35.8801	−1.35324
$704$	0	0
$705$	7.63362	0.287499
$706$	0	0
$707$	6.86392	0.258144
$708$	0	0
$709$	1.23030	0.0462048	0.0231024	−	0.999733i	$-0.492646\pi$
$709$	1.23030	0.0462048	0.0231024	+	0.999733i	$0.492646\pi$
$710$	0	0
$711$	−38.7282	−1.45242
$712$	0	0
$713$	1.22239	0.0457790
$714$	0	0
$715$	−3.81681	−0.142741
$716$	0	0
$717$	27.0083	1.00864
$718$	0	0
$719$	−7.82738	−0.291912	−0.145956	−	0.989291i	$-0.546626\pi$
$719$	−7.82738	−0.291912	−0.145956	+	0.989291i	$0.546626\pi$
$720$	0	0
$721$	5.01621	0.186813
$722$	0	0
$723$	−69.5922	−2.58816
$724$	0	0
$725$	2.85601	0.106070
$726$	0	0
$727$	27.2857	1.01197	0.505986	−	0.862542i	$-0.331129\pi$
$727$	27.2857	1.01197	0.505986	+	0.862542i	$0.331129\pi$
$728$	0	0
$729$	−42.1685	−1.56180
$730$	0	0
$731$	−19.6336	−0.726176
$732$	0	0
$733$	−10.0000	−0.369358	−0.184679	−	0.982799i	$-0.559125\pi$
$733$	−10.0000	−0.369358	−0.184679	+	0.982799i	$0.559125\pi$
$734$	0	0
$735$	−15.2118	−0.561097
$736$	0	0
$737$	−9.50246	−0.350028
$738$	0	0
$739$	−30.5450	−1.12362	−0.561809	−	0.827267i	$-0.689894\pi$
$739$	−30.5450	−1.12362	−0.561809	+	0.827267i	$0.689894\pi$
$740$	0	0
$741$	−14.7776	−0.542869
$742$	0	0
$743$	36.4112	1.33580	0.667899	−	0.744252i	$-0.267194\pi$
$743$	36.4112	1.33580	0.667899	+	0.744252i	$0.267194\pi$
$744$	0	0
$745$	−14.4033	−0.527697
$746$	0	0
$747$	−49.0452	−1.79447
$748$	0	0
$749$	−0.440264	−0.0160869
$750$	0	0
$751$	13.1730	0.480690	0.240345	−	0.970687i	$-0.422739\pi$
$751$	13.1730	0.480690	0.240345	+	0.970687i	$0.422739\pi$
$752$	0	0
$753$	−13.2100	−0.481398
$754$	0	0
$755$	22.5081	0.819154
$756$	0	0
$757$	33.8801	1.23139	0.615697	−	0.787983i	$-0.288874\pi$
$757$	33.8801	1.23139	0.615697	+	0.787983i	$0.288874\pi$
$758$	0	0
$759$	−68.0739	−2.47093
$760$	0	0
$761$	33.2672	1.20594	0.602968	−	0.797765i	$-0.293985\pi$
$761$	33.2672	1.20594	0.602968	+	0.797765i	$0.293985\pi$
$762$	0	0
$763$	−14.5187	−0.525611
$764$	0	0
$765$	30.4403	1.10057
$766$	0	0
$767$	−5.16246	−0.186406
$768$	0	0
$769$	29.8432	1.07617	0.538086	−	0.842890i	$-0.319147\pi$
$769$	29.8432	1.07617	0.538086	+	0.842890i	$0.319147\pi$
$770$	0	0
$771$	−32.8436	−1.18283
$772$	0	0
$773$	−16.7776	−0.603449	−0.301724	−	0.953395i	$-0.597562\pi$
$773$	−16.7776	−0.603449	−0.301724	+	0.953395i	$0.597562\pi$
$774$	0	0
$775$	0.183190	0.00658037
$776$	0	0
$777$	19.8432	0.711870
$778$	0	0
$779$	40.6129	1.45511
$780$	0	0
$781$	14.5680	0.521285
$782$	0	0
$783$	8.73276	0.312083
$784$	0	0
$785$	16.6913	0.595738
$786$	0	0
$787$	−20.1647	−0.718795	−0.359397	−	0.933185i	$-0.617018\pi$
$787$	−20.1647	−0.718795	−0.359397	+	0.933185i	$0.617018\pi$
$788$	0	0
$789$	19.8926	0.708194
$790$	0	0
$791$	1.20997	0.0430214
$792$	0	0
$793$	5.14399	0.182668
$794$	0	0
$795$	−36.4403	−1.29240
$796$	0	0
$797$	3.30871	0.117200	0.0586002	−	0.998282i	$-0.481336\pi$
$797$	3.30871	0.117200	0.0586002	+	0.998282i	$0.481336\pi$
$798$	0	0
$799$	20.9793	0.742193
$800$	0	0
$801$	−62.0739	−2.19327
$802$	0	0
$803$	−48.7697	−1.72105
$804$	0	0
$805$	7.63362	0.269050
$806$	0	0
$807$	−56.8436	−2.00099
$808$	0	0
$809$	22.6834	0.797505	0.398753	−	0.917058i	$-0.369443\pi$
$809$	22.6834	0.797505	0.398753	+	0.917058i	$0.369443\pi$
$810$	0	0
$811$	9.93216	0.348765	0.174383	−	0.984678i	$-0.444207\pi$
$811$	9.93216	0.348765	0.174383	+	0.984678i	$0.444207\pi$
$812$	0	0
$813$	52.9668	1.85763
$814$	0	0
$815$	23.4689	0.822080
$816$	0	0
$817$	14.7776	0.517003
$818$	0	0
$819$	4.74066	0.165652
$820$	0	0
$821$	−38.4033	−1.34029	−0.670143	−	0.742232i	$-0.733767\pi$
$821$	−38.4033	−1.34029	−0.670143	+	0.742232i	$0.733767\pi$
$822$	0	0
$823$	13.1176	0.457251	0.228626	−	0.973514i	$-0.426577\pi$
$823$	13.1176	0.457251	0.228626	+	0.973514i	$0.426577\pi$
$824$	0	0
$825$	−10.2017	−0.355176
$826$	0	0
$827$	16.9714	0.590152	0.295076	−	0.955474i	$-0.404655\pi$
$827$	16.9714	0.590152	0.295076	+	0.955474i	$0.404655\pi$
$828$	0	0
$829$	31.8353	1.10569	0.552843	−	0.833286i	$-0.313543\pi$
$829$	31.8353	1.10569	0.552843	+	0.833286i	$0.313543\pi$
$830$	0	0
$831$	−31.8643	−1.10536
$832$	0	0
$833$	−41.8062	−1.44850
$834$	0	0
$835$	9.14399	0.316441
$836$	0	0
$837$	0.560135	0.0193611
$838$	0	0
$839$	3.62306	0.125082	0.0625409	−	0.998042i	$-0.480080\pi$
$839$	3.62306	0.125082	0.0625409	+	0.998042i	$0.480080\pi$
$840$	0	0
$841$	−20.8432	−0.718731
$842$	0	0
$843$	51.7075	1.78090
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	4.08179	0.140252
$848$	0	0
$849$	−78.4851	−2.69360
$850$	0	0
$851$	43.3042	1.48445
$852$	0	0
$853$	21.5104	0.736501	0.368250	−	0.929727i	$-0.379957\pi$
$853$	21.5104	0.736501	0.368250	+	0.929727i	$0.379957\pi$
$854$	0	0
$855$	−22.9114	−0.783554
$856$	0	0
$857$	−17.6706	−0.603615	−0.301807	−	0.953369i	$-0.597590\pi$
$857$	−17.6706	−0.603615	−0.301807	+	0.953369i	$0.597590\pi$
$858$	0	0
$859$	−3.63362	−0.123978	−0.0619888	−	0.998077i	$-0.519744\pi$
$859$	−3.63362	−0.123978	−0.0619888	+	0.998077i	$0.519744\pi$
$860$	0	0
$861$	−22.4606	−0.765455
$862$	0	0
$863$	13.9506	0.474885	0.237442	−	0.971402i	$-0.423691\pi$
$863$	13.9506	0.474885	0.237442	+	0.971402i	$0.423691\pi$
$864$	0	0
$865$	−10.0369	−0.341266
$866$	0	0
$867$	98.7837	3.35487
$868$	0	0
$869$	35.6706	1.21004
$870$	0	0
$871$	2.48963	0.0843580
$872$	0	0
$873$	31.9585	1.08163
$874$	0	0
$875$	1.14399	0.0386738
$876$	0	0
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	0	0
$879$	−65.7648	−2.21819
$880$	0	0
$881$	−16.3954	−0.552376	−0.276188	−	0.961104i	$-0.589071\pi$
$881$	−16.3954	−0.552376	−0.276188	+	0.961104i	$0.589071\pi$
$882$	0	0
$883$	−44.6314	−1.50196	−0.750982	−	0.660322i	$-0.770420\pi$
$883$	−44.6314	−1.50196	−0.750982	+	0.660322i	$0.770420\pi$
$884$	0	0
$885$	−13.7983	−0.463826
$886$	0	0
$887$	47.4795	1.59420	0.797102	−	0.603844i	$-0.206365\pi$
$887$	47.4795	1.59420	0.797102	+	0.603844i	$0.206365\pi$
$888$	0	0
$889$	−10.2510	−0.343809
$890$	0	0
$891$	16.2571	0.544632
$892$	0	0
$893$	−15.7904	−0.528407
$894$	0	0
$895$	20.9793	0.701260
$896$	0	0
$897$	17.8353	0.595503
$898$	0	0
$899$	0.523192	0.0174494
$900$	0	0
$901$	−100.148	−3.33641
$902$	0	0
$903$	−8.17262	−0.271968
$904$	0	0
$905$	13.5473	0.450328
$906$	0	0
$907$	−22.8824	−0.759797	−0.379899	−	0.925028i	$-0.624041\pi$
$907$	−22.8824	−0.759797	−0.379899	+	0.925028i	$0.624041\pi$
$908$	0	0
$909$	24.8639	0.824684
$910$	0	0
$911$	−0.329437	−0.0109147	−0.00545737	−	0.999985i	$-0.501737\pi$
$911$	−0.329437	−0.0109147	−0.00545737	+	0.999985i	$0.501737\pi$
$912$	0	0
$913$	45.1730	1.49501
$914$	0	0
$915$	13.7490	0.454527
$916$	0	0
$917$	10.2722	0.339217
$918$	0	0
$919$	−36.0369	−1.18875	−0.594375	−	0.804188i	$-0.702600\pi$
$919$	−36.0369	−1.18875	−0.594375	+	0.804188i	$0.702600\pi$
$920$	0	0
$921$	40.7855	1.34393
$922$	0	0
$923$	−3.81681	−0.125632
$924$	0	0
$925$	6.48963	0.213378
$926$	0	0
$927$	18.1708	0.596806
$928$	0	0
$929$	26.7855	0.878804	0.439402	−	0.898290i	$-0.355190\pi$
$929$	26.7855	0.878804	0.439402	+	0.898290i	$0.355190\pi$
$930$	0	0
$931$	31.4662	1.03126
$932$	0	0
$933$	−42.8639	−1.40330
$934$	0	0
$935$	−28.0369	−0.916906
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	74.6295	2.43544
$940$	0	0
$941$	−13.9631	−0.455183	−0.227591	−	0.973757i	$-0.573085\pi$
$941$	−13.9631	−0.455183	−0.227591	+	0.973757i	$0.573085\pi$
$942$	0	0
$943$	−49.0162	−1.59619
$944$	0	0
$945$	3.49794	0.113788
$946$	0	0
$947$	18.1233	0.588927	0.294463	−	0.955663i	$-0.404859\pi$
$947$	18.1233	0.588927	0.294463	+	0.955663i	$0.404859\pi$
$948$	0	0
$949$	12.7776	0.414779
$950$	0	0
$951$	−58.5714	−1.89931
$952$	0	0
$953$	2.97927	0.0965080	0.0482540	−	0.998835i	$-0.484634\pi$
$953$	2.97927	0.0965080	0.0482540	+	0.998835i	$0.484634\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	0	0
$957$	−29.1361	−0.941836
$958$	0	0
$959$	−16.4772	−0.532077
$960$	0	0
$961$	−30.9664	−0.998917
$962$	0	0
$963$	−1.59482	−0.0513922
$964$	0	0
$965$	−3.71203	−0.119494
$966$	0	0
$967$	−6.48963	−0.208693	−0.104346	−	0.994541i	$-0.533275\pi$
$967$	−6.48963	−0.208693	−0.104346	+	0.994541i	$0.533275\pi$
$968$	0	0
$969$	−108.551	−3.48716
$970$	0	0
$971$	−32.9793	−1.05836	−0.529178	−	0.848511i	$-0.677499\pi$
$971$	−32.9793	−1.05836	−0.529178	+	0.848511i	$0.677499\pi$
$972$	0	0
$973$	−24.4191	−0.782841
$974$	0	0
$975$	2.67282	0.0855989
$976$	0	0
$977$	−31.7983	−1.01732	−0.508659	−	0.860968i	$-0.669859\pi$
$977$	−31.7983	−1.01732	−0.508659	+	0.860968i	$0.669859\pi$
$978$	0	0
$979$	57.1730	1.82726
$980$	0	0
$981$	−52.5926	−1.67915
$982$	0	0
$983$	−23.0656	−0.735678	−0.367839	−	0.929890i	$-0.619902\pi$
$983$	−23.0656	−0.735678	−0.367839	+	0.929890i	$0.619902\pi$
$984$	0	0
$985$	8.69129	0.276928
$986$	0	0
$987$	8.73276	0.277967
$988$	0	0
$989$	−17.8353	−0.567129
$990$	0	0
$991$	−52.0739	−1.65418	−0.827091	−	0.562068i	$-0.810006\pi$
$991$	−52.0739	−1.65418	−0.827091	+	0.562068i	$0.810006\pi$
$992$	0	0
$993$	19.3536	0.614166
$994$	0	0
$995$	24.9793	0.791896
$996$	0	0
$997$	−6.78551	−0.214899	−0.107450	−	0.994211i	$-0.534268\pi$
$997$	−6.78551	−0.214899	−0.107450	+	0.994211i	$0.534268\pi$
$998$	0	0
$999$	19.8432	0.627811

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.a.o.1.3		3
3.2	odd	2		9360.2.a.da.1.2		3
4.3	odd	2		260.2.a.b.1.1	✓	3
5.4	even	2		5200.2.a.ci.1.1		3
8.3	odd	2		4160.2.a.bo.1.3		3
8.5	even	2		4160.2.a.br.1.1		3
12.11	even	2		2340.2.a.n.1.2		3
20.3	even	4		1300.2.c.f.1249.2		6
20.7	even	4		1300.2.c.f.1249.5		6
20.19	odd	2		1300.2.a.i.1.3		3
52.31	even	4		3380.2.f.h.3041.1		6
52.47	even	4		3380.2.f.h.3041.2		6
52.51	odd	2		3380.2.a.o.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.a.b.1.1	✓	3	4.3	odd	2
1040.2.a.o.1.3		3	1.1	even	1	trivial
1300.2.a.i.1.3		3	20.19	odd	2
1300.2.c.f.1249.2		6	20.3	even	4
1300.2.c.f.1249.5		6	20.7	even	4
2340.2.a.n.1.2		3	12.11	even	2
3380.2.a.o.1.1		3	52.51	odd	2
3380.2.f.h.3041.1		6	52.31	even	4
3380.2.f.h.3041.2		6	52.47	even	4
4160.2.a.bo.1.3		3	8.3	odd	2
4160.2.a.br.1.1		3	8.5	even	2
5200.2.a.ci.1.1		3	5.4	even	2
9360.2.a.da.1.2		3	3.2	odd	2

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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

\(f(q)\)	\(=\)	\(q+2.67282 q^{3} +1.00000 q^{5} +1.14399 q^{7} +4.14399 q^{9} +O(q^{10})\)	\(q+2.67282 q^{3} +1.00000 q^{5} +1.14399 q^{7} +4.14399 q^{9} -3.81681 q^{11} +1.00000 q^{13} +2.67282 q^{15} +7.34565 q^{17} -5.52884 q^{19} +3.05767 q^{21} +6.67282 q^{23} +1.00000 q^{25} +3.05767 q^{27} +2.85601 q^{29} +0.183190 q^{31} -10.2017 q^{33} +1.14399 q^{35} +6.48963 q^{37} +2.67282 q^{39} -7.34565 q^{41} -2.67282 q^{43} +4.14399 q^{45} +2.85601 q^{47} -5.69129 q^{49} +19.6336 q^{51} -13.6336 q^{53} -3.81681 q^{55} -14.7776 q^{57} -5.16246 q^{59} +5.14399 q^{61} +4.74066 q^{63} +1.00000 q^{65} +2.48963 q^{67} +17.8353 q^{69} -3.81681 q^{71} +12.7776 q^{73} +2.67282 q^{75} -4.36638 q^{77} -9.34565 q^{79} -4.25934 q^{81} -11.8353 q^{83} +7.34565 q^{85} +7.63362 q^{87} -14.9793 q^{89} +1.14399 q^{91} +0.489634 q^{93} -5.52884 q^{95} +7.71203 q^{97} -15.8168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 3 q^{5} + 2 q^{7} + 11 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 + 3 * q^5 + 2 * q^7 + 11 * q^9	\( 3 q - 2 q^{3} + 3 q^{5} + 2 q^{7} + 11 q^{9} + 3 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} - 8 q^{21} + 10 q^{23} + 3 q^{25} - 8 q^{27} + 10 q^{29} + 12 q^{31} - 12 q^{33} + 2 q^{35} - 2 q^{37} - 2 q^{39} - 2 q^{41} + 2 q^{43} + 11 q^{45} + 10 q^{47} + 23 q^{49} + 36 q^{51} - 18 q^{53} - 20 q^{57} + 16 q^{59} + 14 q^{61} + 50 q^{63} + 3 q^{65} - 14 q^{67} + 12 q^{69} + 14 q^{73} - 2 q^{75} - 36 q^{77} - 8 q^{79} + 23 q^{81} + 6 q^{83} + 2 q^{85} - 2 q^{89} + 2 q^{91} - 20 q^{93} - 8 q^{95} + 26 q^{97} - 36 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 3 * q^5 + 2 * q^7 + 11 * q^9 + 3 * q^13 - 2 * q^15 + 2 * q^17 - 8 * q^19 - 8 * q^21 + 10 * q^23 + 3 * q^25 - 8 * q^27 + 10 * q^29 + 12 * q^31 - 12 * q^33 + 2 * q^35 - 2 * q^37 - 2 * q^39 - 2 * q^41 + 2 * q^43 + 11 * q^45 + 10 * q^47 + 23 * q^49 + 36 * q^51 - 18 * q^53 - 20 * q^57 + 16 * q^59 + 14 * q^61 + 50 * q^63 + 3 * q^65 - 14 * q^67 + 12 * q^69 + 14 * q^73 - 2 * q^75 - 36 * q^77 - 8 * q^79 + 23 * q^81 + 6 * q^83 + 2 * q^85 - 2 * q^89 + 2 * q^91 - 20 * q^93 - 8 * q^95 + 26 * q^97 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more