LMFDB - Embedded newform 3640.2.a.z.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2112217.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 10x^{3} - x^{2} + 10x + 4 $ x^5 - 10x^3 - x^2 + 10x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.492015$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.492015 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.75792 q^{9} +O(q^{10})$	$q+0.492015 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.75792 q^{9} +6.19270 q^{11} -1.00000 q^{13} +0.492015 q^{15} -1.82079 q^{17} +3.32500 q^{19} +0.492015 q^{21} -3.20867 q^{23} +1.00000 q^{25} -2.83299 q^{27} +4.71665 q^{29} +2.49202 q^{31} +3.04690 q^{33} +1.00000 q^{35} -3.63781 q^{37} -0.492015 q^{39} +2.77389 q^{41} +10.7874 q^{43} -2.75792 q^{45} +10.3225 q^{47} +1.00000 q^{49} -0.895858 q^{51} -3.40136 q^{53} +6.19270 q^{55} +1.63595 q^{57} -9.83051 q^{59} +4.92116 q^{61} -2.75792 q^{63} -1.00000 q^{65} -14.8009 q^{67} -1.57871 q^{69} +9.89716 q^{71} +2.72043 q^{73} +0.492015 q^{75} +6.19270 q^{77} +12.1433 q^{79} +6.87989 q^{81} -7.52711 q^{83} -1.82079 q^{85} +2.32067 q^{87} +2.83113 q^{89} -1.00000 q^{91} +1.22611 q^{93} +3.32500 q^{95} -17.8816 q^{97} -17.0790 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9	$ 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9} + 7 q^{11} - 5 q^{13} + 3 q^{17} + 3 q^{19} + 3 q^{23} + 5 q^{25} - 3 q^{27} + 7 q^{29} + 10 q^{31} + 17 q^{33} + 5 q^{35} + 10 q^{37} + 4 q^{43} + 5 q^{45} - 3 q^{47} + 5 q^{49} + 26 q^{53} + 7 q^{55} - 20 q^{57} + 3 q^{59} + 13 q^{61} + 5 q^{63} - 5 q^{65} + 12 q^{67} + 23 q^{69} + 8 q^{71} + q^{73} + 7 q^{77} - 6 q^{79} + 25 q^{81} - 8 q^{83} + 3 q^{85} - 43 q^{87} + 3 q^{89} - 5 q^{91} + 20 q^{93} + 3 q^{95} - 15 q^{97} + 16 q^{99}+O(q^{100}) $ 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9 + 7 * q^11 - 5 * q^13 + 3 * q^17 + 3 * q^19 + 3 * q^23 + 5 * q^25 - 3 * q^27 + 7 * q^29 + 10 * q^31 + 17 * q^33 + 5 * q^35 + 10 * q^37 + 4 * q^43 + 5 * q^45 - 3 * q^47 + 5 * q^49 + 26 * q^53 + 7 * q^55 - 20 * q^57 + 3 * q^59 + 13 * q^61 + 5 * q^63 - 5 * q^65 + 12 * q^67 + 23 * q^69 + 8 * q^71 + q^73 + 7 * q^77 - 6 * q^79 + 25 * q^81 - 8 * q^83 + 3 * q^85 - 43 * q^87 + 3 * q^89 - 5 * q^91 + 20 * q^93 + 3 * q^95 - 15 * q^97 + 16 * 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
$3$	0.492015	0.284065	0.142033	−	0.989862i	$-0.454636\pi$
$3$	0.492015	0.284065	0.142033	+	0.989862i	$0.454636\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.75792	−0.919307
$10$	0	0
$11$	6.19270	1.86717	0.933584	−	0.358358i	$-0.116663\pi$
$11$	6.19270	1.86717	0.933584	+	0.358358i	$0.116663\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0.492015	0.127038
$16$	0	0
$17$	−1.82079	−0.441607	−0.220804	−	0.975318i	$-0.570868\pi$
$17$	−1.82079	−0.441607	−0.220804	+	0.975318i	$0.570868\pi$
$18$	0	0
$19$	3.32500	0.762808	0.381404	−	0.924409i	$-0.375441\pi$
$19$	3.32500	0.762808	0.381404	+	0.924409i	$0.375441\pi$
$20$	0	0
$21$	0.492015	0.107367
$22$	0	0
$23$	−3.20867	−0.669053	−0.334527	−	0.942386i	$-0.608576\pi$
$23$	−3.20867	−0.669053	−0.334527	+	0.942386i	$0.608576\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.83299	−0.545208
$28$	0	0
$29$	4.71665	0.875860	0.437930	−	0.899009i	$-0.355712\pi$
$29$	4.71665	0.875860	0.437930	+	0.899009i	$0.355712\pi$
$30$	0	0
$31$	2.49202	0.447579	0.223790	−	0.974637i	$-0.428157\pi$
$31$	2.49202	0.447579	0.223790	+	0.974637i	$0.428157\pi$
$32$	0	0
$33$	3.04690	0.530398
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−3.63781	−0.598052	−0.299026	−	0.954245i	$-0.596662\pi$
$37$	−3.63781	−0.598052	−0.299026	+	0.954245i	$0.596662\pi$
$38$	0	0
$39$	−0.492015	−0.0787855
$40$	0	0
$41$	2.77389	0.433209	0.216604	−	0.976259i	$-0.430502\pi$
$41$	2.77389	0.433209	0.216604	+	0.976259i	$0.430502\pi$
$42$	0	0
$43$	10.7874	1.64506	0.822530	−	0.568722i	$-0.192562\pi$
$43$	10.7874	1.64506	0.822530	+	0.568722i	$0.192562\pi$
$44$	0	0
$45$	−2.75792	−0.411127
$46$	0	0
$47$	10.3225	1.50569	0.752847	−	0.658195i	$-0.228680\pi$
$47$	10.3225	1.50569	0.752847	+	0.658195i	$0.228680\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.895858	−0.125445
$52$	0	0
$53$	−3.40136	−0.467213	−0.233607	−	0.972331i	$-0.575053\pi$
$53$	−3.40136	−0.467213	−0.233607	+	0.972331i	$0.575053\pi$
$54$	0	0
$55$	6.19270	0.835023
$56$	0	0
$57$	1.63595	0.216687
$58$	0	0
$59$	−9.83051	−1.27982	−0.639912	−	0.768449i	$-0.721029\pi$
$59$	−9.83051	−1.27982	−0.639912	+	0.768449i	$0.721029\pi$
$60$	0	0
$61$	4.92116	0.630090	0.315045	−	0.949077i	$-0.397980\pi$
$61$	4.92116	0.630090	0.315045	+	0.949077i	$0.397980\pi$
$62$	0	0
$63$	−2.75792	−0.347465
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−14.8009	−1.80821	−0.904107	−	0.427306i	$-0.859463\pi$
$67$	−14.8009	−1.80821	−0.904107	+	0.427306i	$0.859463\pi$
$68$	0	0
$69$	−1.57871	−0.190055
$70$	0	0
$71$	9.89716	1.17458	0.587288	−	0.809378i	$-0.300196\pi$
$71$	9.89716	1.17458	0.587288	+	0.809378i	$0.300196\pi$
$72$	0	0
$73$	2.72043	0.318402	0.159201	−	0.987246i	$-0.449108\pi$
$73$	2.72043	0.318402	0.159201	+	0.987246i	$0.449108\pi$
$74$	0	0
$75$	0.492015	0.0568131
$76$	0	0
$77$	6.19270	0.705723
$78$	0	0
$79$	12.1433	1.36623	0.683115	−	0.730311i	$-0.260625\pi$
$79$	12.1433	1.36623	0.683115	+	0.730311i	$0.260625\pi$
$80$	0	0
$81$	6.87989	0.764432
$82$	0	0
$83$	−7.52711	−0.826208	−0.413104	−	0.910684i	$-0.635555\pi$
$83$	−7.52711	−0.826208	−0.413104	+	0.910684i	$0.635555\pi$
$84$	0	0
$85$	−1.82079	−0.197493
$86$	0	0
$87$	2.32067	0.248801
$88$	0	0
$89$	2.83113	0.300099	0.150050	−	0.988678i	$-0.452057\pi$
$89$	2.83113	0.300099	0.150050	+	0.988678i	$0.452057\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	1.22611	0.127142
$94$	0	0
$95$	3.32500	0.341138
$96$	0	0
$97$	−17.8816	−1.81560	−0.907799	−	0.419405i	$-0.862239\pi$
$97$	−17.8816	−1.81560	−0.907799	+	0.419405i	$0.862239\pi$
$98$	0	0
$99$	−17.0790	−1.71650
$100$	0	0
$101$	−1.67252	−0.166422	−0.0832110	−	0.996532i	$-0.526518\pi$
$101$	−1.67252	−0.166422	−0.0832110	+	0.996532i	$0.526518\pi$
$102$	0	0
$103$	8.84084	0.871114	0.435557	−	0.900161i	$-0.356551\pi$
$103$	8.84084	0.871114	0.435557	+	0.900161i	$0.356551\pi$
$104$	0	0
$105$	0.492015	0.0480158
$106$	0	0
$107$	−4.49579	−0.434625	−0.217312	−	0.976102i	$-0.569729\pi$
$107$	−4.49579	−0.434625	−0.217312	+	0.976102i	$0.569729\pi$
$108$	0	0
$109$	4.12574	0.395175	0.197587	−	0.980285i	$-0.436689\pi$
$109$	4.12574	0.395175	0.197587	+	0.980285i	$0.436689\pi$
$110$	0	0
$111$	−1.78986	−0.169886
$112$	0	0
$113$	14.2287	1.33853	0.669263	−	0.743026i	$-0.266610\pi$
$113$	14.2287	1.33853	0.669263	+	0.743026i	$0.266610\pi$
$114$	0	0
$115$	−3.20867	−0.299210
$116$	0	0
$117$	2.75792	0.254970
$118$	0	0
$119$	−1.82079	−0.166912
$120$	0	0
$121$	27.3495	2.48632
$122$	0	0
$123$	1.36480	0.123060
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.30717	−0.382200	−0.191100	−	0.981571i	$-0.561205\pi$
$127$	−4.30717	−0.382200	−0.191100	+	0.981571i	$0.561205\pi$
$128$	0	0
$129$	5.30756	0.467304
$130$	0	0
$131$	11.2832	0.985815	0.492908	−	0.870082i	$-0.335934\pi$
$131$	11.2832	0.985815	0.492908	+	0.870082i	$0.335934\pi$
$132$	0	0
$133$	3.32500	0.288314
$134$	0	0
$135$	−2.83299	−0.243825
$136$	0	0
$137$	13.4680	1.15065	0.575325	−	0.817925i	$-0.304876\pi$
$137$	13.4680	1.15065	0.575325	+	0.817925i	$0.304876\pi$
$138$	0	0
$139$	−7.75978	−0.658176	−0.329088	−	0.944299i	$-0.606741\pi$
$139$	−7.75978	−0.658176	−0.329088	+	0.944299i	$0.606741\pi$
$140$	0	0
$141$	5.07884	0.427716
$142$	0	0
$143$	−6.19270	−0.517859
$144$	0	0
$145$	4.71665	0.391697
$146$	0	0
$147$	0.492015	0.0405808
$148$	0	0
$149$	0.838620	0.0687024	0.0343512	−	0.999410i	$-0.489064\pi$
$149$	0.838620	0.0687024	0.0343512	+	0.999410i	$0.489064\pi$
$150$	0	0
$151$	−13.4130	−1.09153	−0.545767	−	0.837937i	$-0.683762\pi$
$151$	−13.4130	−1.09153	−0.545767	+	0.837937i	$0.683762\pi$
$152$	0	0
$153$	5.02160	0.405973
$154$	0	0
$155$	2.49202	0.200164
$156$	0	0
$157$	19.1649	1.52953	0.764764	−	0.644311i	$-0.222856\pi$
$157$	19.1649	1.52953	0.764764	+	0.644311i	$0.222856\pi$
$158$	0	0
$159$	−1.67352	−0.132719
$160$	0	0
$161$	−3.20867	−0.252878
$162$	0	0
$163$	15.2278	1.19273	0.596366	−	0.802712i	$-0.296611\pi$
$163$	15.2278	1.19273	0.596366	+	0.802712i	$0.296611\pi$
$164$	0	0
$165$	3.04690	0.237201
$166$	0	0
$167$	−8.56375	−0.662683	−0.331341	−	0.943511i	$-0.607501\pi$
$167$	−8.56375	−0.662683	−0.331341	+	0.943511i	$0.607501\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−9.17009	−0.701254
$172$	0	0
$173$	2.92493	0.222379	0.111189	−	0.993799i	$-0.464534\pi$
$173$	2.92493	0.222379	0.111189	+	0.993799i	$0.464534\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−4.83676	−0.363553
$178$	0	0
$179$	−20.6047	−1.54007	−0.770034	−	0.638003i	$-0.779761\pi$
$179$	−20.6047	−1.54007	−0.770034	+	0.638003i	$0.779761\pi$
$180$	0	0
$181$	14.4689	1.07547	0.537734	−	0.843115i	$-0.319281\pi$
$181$	14.4689	1.07547	0.537734	+	0.843115i	$0.319281\pi$
$182$	0	0
$183$	2.42129	0.178987
$184$	0	0
$185$	−3.63781	−0.267457
$186$	0	0
$187$	−11.2756	−0.824555
$188$	0	0
$189$	−2.83299	−0.206069
$190$	0	0
$191$	−18.6179	−1.34714	−0.673572	−	0.739122i	$-0.735241\pi$
$191$	−18.6179	−1.34714	−0.673572	+	0.739122i	$0.735241\pi$
$192$	0	0
$193$	14.2269	1.02407	0.512036	−	0.858964i	$-0.328891\pi$
$193$	14.2269	1.02407	0.512036	+	0.858964i	$0.328891\pi$
$194$	0	0
$195$	−0.492015	−0.0352340
$196$	0	0
$197$	−9.25872	−0.659657	−0.329828	−	0.944041i	$-0.606991\pi$
$197$	−9.25872	−0.659657	−0.329828	+	0.944041i	$0.606991\pi$
$198$	0	0
$199$	−19.6686	−1.39427	−0.697134	−	0.716941i	$-0.745542\pi$
$199$	−19.6686	−1.39427	−0.697134	+	0.716941i	$0.745542\pi$
$200$	0	0
$201$	−7.28226	−0.513651
$202$	0	0
$203$	4.71665	0.331044
$204$	0	0
$205$	2.77389	0.193737
$206$	0	0
$207$	8.84925	0.615065
$208$	0	0
$209$	20.5907	1.42429
$210$	0	0
$211$	23.7540	1.63529	0.817645	−	0.575722i	$-0.195279\pi$
$211$	23.7540	1.63529	0.817645	+	0.575722i	$0.195279\pi$
$212$	0	0
$213$	4.86955	0.333656
$214$	0	0
$215$	10.7874	0.735693
$216$	0	0
$217$	2.49202	0.169169
$218$	0	0
$219$	1.33849	0.0904469
$220$	0	0
$221$	1.82079	0.122480
$222$	0	0
$223$	14.4134	0.965191	0.482596	−	0.875843i	$-0.339694\pi$
$223$	14.4134	0.965191	0.482596	+	0.875843i	$0.339694\pi$
$224$	0	0
$225$	−2.75792	−0.183861
$226$	0	0
$227$	22.0783	1.46539	0.732696	−	0.680556i	$-0.238262\pi$
$227$	22.0783	1.46539	0.732696	+	0.680556i	$0.238262\pi$
$228$	0	0
$229$	−24.5728	−1.62381	−0.811907	−	0.583787i	$-0.801570\pi$
$229$	−24.5728	−1.62381	−0.811907	+	0.583787i	$0.801570\pi$
$230$	0	0
$231$	3.04690	0.200472
$232$	0	0
$233$	−19.2397	−1.26044	−0.630218	−	0.776418i	$-0.717035\pi$
$233$	−19.2397	−1.26044	−0.630218	+	0.776418i	$0.717035\pi$
$234$	0	0
$235$	10.3225	0.673367
$236$	0	0
$237$	5.97470	0.388098
$238$	0	0
$239$	22.4534	1.45239	0.726193	−	0.687491i	$-0.241288\pi$
$239$	22.4534	1.45239	0.726193	+	0.687491i	$0.241288\pi$
$240$	0	0
$241$	7.90705	0.509338	0.254669	−	0.967028i	$-0.418034\pi$
$241$	7.90705	0.509338	0.254669	+	0.967028i	$0.418034\pi$
$242$	0	0
$243$	11.8840	0.762357
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−3.32500	−0.211565
$248$	0	0
$249$	−3.70345	−0.234697
$250$	0	0
$251$	2.29554	0.144893	0.0724467	−	0.997372i	$-0.476919\pi$
$251$	2.29554	0.144893	0.0724467	+	0.997372i	$0.476919\pi$
$252$	0	0
$253$	−19.8703	−1.24924
$254$	0	0
$255$	−0.895858	−0.0561008
$256$	0	0
$257$	6.44467	0.402007	0.201004	−	0.979590i	$-0.435580\pi$
$257$	6.44467	0.402007	0.201004	+	0.979590i	$0.435580\pi$
$258$	0	0
$259$	−3.63781	−0.226043
$260$	0	0
$261$	−13.0082	−0.805184
$262$	0	0
$263$	−24.2431	−1.49489	−0.747446	−	0.664323i	$-0.768720\pi$
$263$	−24.2431	−1.49489	−0.747446	+	0.664323i	$0.768720\pi$
$264$	0	0
$265$	−3.40136	−0.208944
$266$	0	0
$267$	1.39296	0.0852477
$268$	0	0
$269$	−9.66497	−0.589284	−0.294642	−	0.955608i	$-0.595200\pi$
$269$	−9.66497	−0.589284	−0.294642	+	0.955608i	$0.595200\pi$
$270$	0	0
$271$	−18.2197	−1.10677	−0.553383	−	0.832927i	$-0.686664\pi$
$271$	−18.2197	−1.10677	−0.553383	+	0.832927i	$0.686664\pi$
$272$	0	0
$273$	−0.492015	−0.0297781
$274$	0	0
$275$	6.19270	0.373434
$276$	0	0
$277$	19.2193	1.15478	0.577388	−	0.816470i	$-0.304072\pi$
$277$	19.2193	1.15478	0.577388	+	0.816470i	$0.304072\pi$
$278$	0	0
$279$	−6.87278	−0.411463
$280$	0	0
$281$	−1.69590	−0.101169	−0.0505845	−	0.998720i	$-0.516108\pi$
$281$	−1.69590	−0.101169	−0.0505845	+	0.998720i	$0.516108\pi$
$282$	0	0
$283$	−28.1155	−1.67129	−0.835645	−	0.549269i	$-0.814906\pi$
$283$	−28.1155	−1.67129	−0.835645	+	0.549269i	$0.814906\pi$
$284$	0	0
$285$	1.63595	0.0969054
$286$	0	0
$287$	2.77389	0.163738
$288$	0	0
$289$	−13.6847	−0.804983
$290$	0	0
$291$	−8.79801	−0.515748
$292$	0	0
$293$	10.8620	0.634565	0.317282	−	0.948331i	$-0.397230\pi$
$293$	10.8620	0.634565	0.317282	+	0.948331i	$0.397230\pi$
$294$	0	0
$295$	−9.83051	−0.572354
$296$	0	0
$297$	−17.5438	−1.01800
$298$	0	0
$299$	3.20867	0.185562
$300$	0	0
$301$	10.7874	0.621774
$302$	0	0
$303$	−0.822906	−0.0472747
$304$	0	0
$305$	4.92116	0.281785
$306$	0	0
$307$	13.8693	0.791563	0.395781	−	0.918345i	$-0.370474\pi$
$307$	13.8693	0.791563	0.395781	+	0.918345i	$0.370474\pi$
$308$	0	0
$309$	4.34983	0.247453
$310$	0	0
$311$	32.2248	1.82730	0.913649	−	0.406503i	$-0.133252\pi$
$311$	32.2248	1.82730	0.913649	+	0.406503i	$0.133252\pi$
$312$	0	0
$313$	−6.84240	−0.386755	−0.193378	−	0.981124i	$-0.561944\pi$
$313$	−6.84240	−0.386755	−0.193378	+	0.981124i	$0.561944\pi$
$314$	0	0
$315$	−2.75792	−0.155391
$316$	0	0
$317$	1.03564	0.0581671	0.0290836	−	0.999577i	$-0.490741\pi$
$317$	1.03564	0.0581671	0.0290836	+	0.999577i	$0.490741\pi$
$318$	0	0
$319$	29.2088	1.63538
$320$	0	0
$321$	−2.21200	−0.123462
$322$	0	0
$323$	−6.05414	−0.336861
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	2.02993	0.112255
$328$	0	0
$329$	10.3225	0.569099
$330$	0	0
$331$	−32.0347	−1.76078	−0.880392	−	0.474246i	$-0.842721\pi$
$331$	−32.0347	−1.76078	−0.880392	+	0.474246i	$0.842721\pi$
$332$	0	0
$333$	10.0328	0.549794
$334$	0	0
$335$	−14.8009	−0.808658
$336$	0	0
$337$	14.1218	0.769263	0.384631	−	0.923070i	$-0.374329\pi$
$337$	14.1218	0.769263	0.384631	+	0.923070i	$0.374329\pi$
$338$	0	0
$339$	7.00075	0.380228
$340$	0	0
$341$	15.4323	0.835706
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−1.57871	−0.0849951
$346$	0	0
$347$	−19.5199	−1.04788	−0.523942	−	0.851754i	$-0.675539\pi$
$347$	−19.5199	−1.04788	−0.523942	+	0.851754i	$0.675539\pi$
$348$	0	0
$349$	22.4332	1.20082	0.600411	−	0.799691i	$-0.295003\pi$
$349$	22.4332	1.20082	0.600411	+	0.799691i	$0.295003\pi$
$350$	0	0
$351$	2.83299	0.151214
$352$	0	0
$353$	−29.2641	−1.55757	−0.778786	−	0.627290i	$-0.784164\pi$
$353$	−29.2641	−1.55757	−0.778786	+	0.627290i	$0.784164\pi$
$354$	0	0
$355$	9.89716	0.525286
$356$	0	0
$357$	−0.895858	−0.0474138
$358$	0	0
$359$	24.5103	1.29360	0.646801	−	0.762659i	$-0.276106\pi$
$359$	24.5103	1.29360	0.646801	+	0.762659i	$0.276106\pi$
$360$	0	0
$361$	−7.94436	−0.418124
$362$	0	0
$363$	13.4564	0.706277
$364$	0	0
$365$	2.72043	0.142394
$366$	0	0
$367$	−12.8467	−0.670590	−0.335295	−	0.942113i	$-0.608836\pi$
$367$	−12.8467	−0.670590	−0.335295	+	0.942113i	$0.608836\pi$
$368$	0	0
$369$	−7.65017	−0.398252
$370$	0	0
$371$	−3.40136	−0.176590
$372$	0	0
$373$	23.7519	1.22983	0.614913	−	0.788595i	$-0.289191\pi$
$373$	23.7519	1.22983	0.614913	+	0.788595i	$0.289191\pi$
$374$	0	0
$375$	0.492015	0.0254076
$376$	0	0
$377$	−4.71665	−0.242920
$378$	0	0
$379$	6.97700	0.358384	0.179192	−	0.983814i	$-0.442652\pi$
$379$	6.97700	0.358384	0.179192	+	0.983814i	$0.442652\pi$
$380$	0	0
$381$	−2.11920	−0.108570
$382$	0	0
$383$	−15.1114	−0.772156	−0.386078	−	0.922466i	$-0.626170\pi$
$383$	−15.1114	−0.772156	−0.386078	+	0.922466i	$0.626170\pi$
$384$	0	0
$385$	6.19270	0.315609
$386$	0	0
$387$	−29.7507	−1.51232
$388$	0	0
$389$	10.0215	0.508111	0.254056	−	0.967190i	$-0.418235\pi$
$389$	10.0215	0.508111	0.254056	+	0.967190i	$0.418235\pi$
$390$	0	0
$391$	5.84232	0.295459
$392$	0	0
$393$	5.55150	0.280036
$394$	0	0
$395$	12.1433	0.610997
$396$	0	0
$397$	16.4098	0.823583	0.411792	−	0.911278i	$-0.364903\pi$
$397$	16.4098	0.823583	0.411792	+	0.911278i	$0.364903\pi$
$398$	0	0
$399$	1.63595	0.0819001
$400$	0	0
$401$	21.1048	1.05392	0.526962	−	0.849889i	$-0.323331\pi$
$401$	21.1048	1.05392	0.526962	+	0.849889i	$0.323331\pi$
$402$	0	0
$403$	−2.49202	−0.124136
$404$	0	0
$405$	6.87989	0.341864
$406$	0	0
$407$	−22.5279	−1.11666
$408$	0	0
$409$	−8.36597	−0.413670	−0.206835	−	0.978376i	$-0.566316\pi$
$409$	−8.36597	−0.413670	−0.206835	+	0.978376i	$0.566316\pi$
$410$	0	0
$411$	6.62647	0.326860
$412$	0	0
$413$	−9.83051	−0.483728
$414$	0	0
$415$	−7.52711	−0.369491
$416$	0	0
$417$	−3.81793	−0.186965
$418$	0	0
$419$	9.90224	0.483756	0.241878	−	0.970307i	$-0.422237\pi$
$419$	9.90224	0.483756	0.241878	+	0.970307i	$0.422237\pi$
$420$	0	0
$421$	−4.59840	−0.224112	−0.112056	−	0.993702i	$-0.535744\pi$
$421$	−4.59840	−0.224112	−0.112056	+	0.993702i	$0.535744\pi$
$422$	0	0
$423$	−28.4687	−1.38420
$424$	0	0
$425$	−1.82079	−0.0883214
$426$	0	0
$427$	4.92116	0.238152
$428$	0	0
$429$	−3.04690	−0.147106
$430$	0	0
$431$	21.6026	1.04056	0.520281	−	0.853995i	$-0.325827\pi$
$431$	21.6026	1.04056	0.520281	+	0.853995i	$0.325827\pi$
$432$	0	0
$433$	−28.4359	−1.36654	−0.683271	−	0.730165i	$-0.739443\pi$
$433$	−28.4359	−1.36654	−0.683271	+	0.730165i	$0.739443\pi$
$434$	0	0
$435$	2.32067	0.111267
$436$	0	0
$437$	−10.6688	−0.510359
$438$	0	0
$439$	9.52416	0.454563	0.227282	−	0.973829i	$-0.427016\pi$
$439$	9.52416	0.454563	0.227282	+	0.973829i	$0.427016\pi$
$440$	0	0
$441$	−2.75792	−0.131330
$442$	0	0
$443$	18.7525	0.890958	0.445479	−	0.895292i	$-0.353033\pi$
$443$	18.7525	0.890958	0.445479	+	0.895292i	$0.353033\pi$
$444$	0	0
$445$	2.83113	0.134208
$446$	0	0
$447$	0.412614	0.0195160
$448$	0	0
$449$	−18.0351	−0.851131	−0.425566	−	0.904928i	$-0.639925\pi$
$449$	−18.0351	−0.851131	−0.425566	+	0.904928i	$0.639925\pi$
$450$	0	0
$451$	17.1779	0.808874
$452$	0	0
$453$	−6.59940	−0.310067
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−8.50005	−0.397616	−0.198808	−	0.980039i	$-0.563707\pi$
$457$	−8.50005	−0.397616	−0.198808	+	0.980039i	$0.563707\pi$
$458$	0	0
$459$	5.15828	0.240768
$460$	0	0
$461$	12.6603	0.589650	0.294825	−	0.955551i	$-0.404739\pi$
$461$	12.6603	0.589650	0.294825	+	0.955551i	$0.404739\pi$
$462$	0	0
$463$	−29.8823	−1.38875	−0.694374	−	0.719614i	$-0.744319\pi$
$463$	−29.8823	−1.38875	−0.694374	+	0.719614i	$0.744319\pi$
$464$	0	0
$465$	1.22611	0.0568595
$466$	0	0
$467$	−9.07166	−0.419786	−0.209893	−	0.977724i	$-0.567312\pi$
$467$	−9.07166	−0.419786	−0.209893	+	0.977724i	$0.567312\pi$
$468$	0	0
$469$	−14.8009	−0.683441
$470$	0	0
$471$	9.42944	0.434486
$472$	0	0
$473$	66.8030	3.07160
$474$	0	0
$475$	3.32500	0.152562
$476$	0	0
$477$	9.38069	0.429512
$478$	0	0
$479$	−2.34660	−0.107219	−0.0536095	−	0.998562i	$-0.517073\pi$
$479$	−2.34660	−0.107219	−0.0536095	+	0.998562i	$0.517073\pi$
$480$	0	0
$481$	3.63781	0.165870
$482$	0	0
$483$	−1.57871	−0.0718340
$484$	0	0
$485$	−17.8816	−0.811960
$486$	0	0
$487$	−13.2833	−0.601926	−0.300963	−	0.953636i	$-0.597308\pi$
$487$	−13.2833	−0.601926	−0.300963	+	0.953636i	$0.597308\pi$
$488$	0	0
$489$	7.49231	0.338814
$490$	0	0
$491$	−12.7708	−0.576338	−0.288169	−	0.957580i	$-0.593046\pi$
$491$	−12.7708	−0.576338	−0.288169	+	0.957580i	$0.593046\pi$
$492$	0	0
$493$	−8.58805	−0.386786
$494$	0	0
$495$	−17.0790	−0.767643
$496$	0	0
$497$	9.89716	0.443948
$498$	0	0
$499$	−2.88591	−0.129191	−0.0645955	−	0.997912i	$-0.520576\pi$
$499$	−2.88591	−0.129191	−0.0645955	+	0.997912i	$0.520576\pi$
$500$	0	0
$501$	−4.21350	−0.188245
$502$	0	0
$503$	7.90087	0.352282	0.176141	−	0.984365i	$-0.443638\pi$
$503$	7.90087	0.352282	0.176141	+	0.984365i	$0.443638\pi$
$504$	0	0
$505$	−1.67252	−0.0744262
$506$	0	0
$507$	0.492015	0.0218512
$508$	0	0
$509$	−17.2428	−0.764275	−0.382137	−	0.924105i	$-0.624812\pi$
$509$	−17.2428	−0.764275	−0.382137	+	0.924105i	$0.624812\pi$
$510$	0	0
$511$	2.72043	0.120345
$512$	0	0
$513$	−9.41968	−0.415889
$514$	0	0
$515$	8.84084	0.389574
$516$	0	0
$517$	63.9243	2.81139
$518$	0	0
$519$	1.43911	0.0631701
$520$	0	0
$521$	−35.4873	−1.55473	−0.777363	−	0.629053i	$-0.783443\pi$
$521$	−35.4873	−1.55473	−0.777363	+	0.629053i	$0.783443\pi$
$522$	0	0
$523$	−27.5941	−1.20660	−0.603302	−	0.797513i	$-0.706149\pi$
$523$	−27.5941	−1.20660	−0.603302	+	0.797513i	$0.706149\pi$
$524$	0	0
$525$	0.492015	0.0214733
$526$	0	0
$527$	−4.53744	−0.197654
$528$	0	0
$529$	−12.7045	−0.552368
$530$	0	0
$531$	27.1118	1.17655
$532$	0	0
$533$	−2.77389	−0.120151
$534$	0	0
$535$	−4.49579	−0.194370
$536$	0	0
$537$	−10.1378	−0.437480
$538$	0	0
$539$	6.19270	0.266738
$540$	0	0
$541$	−20.6601	−0.888249	−0.444125	−	0.895965i	$-0.646485\pi$
$541$	−20.6601	−0.888249	−0.444125	+	0.895965i	$0.646485\pi$
$542$	0	0
$543$	7.11894	0.305503
$544$	0	0
$545$	4.12574	0.176727
$546$	0	0
$547$	−14.0866	−0.602300	−0.301150	−	0.953577i	$-0.597371\pi$
$547$	−14.0866	−0.602300	−0.301150	+	0.953577i	$0.597371\pi$
$548$	0	0
$549$	−13.5722	−0.579246
$550$	0	0
$551$	15.6829	0.668113
$552$	0	0
$553$	12.1433	0.516386
$554$	0	0
$555$	−1.78986	−0.0759753
$556$	0	0
$557$	3.87535	0.164204	0.0821019	−	0.996624i	$-0.473837\pi$
$557$	3.87535	0.164204	0.0821019	+	0.996624i	$0.473837\pi$
$558$	0	0
$559$	−10.7874	−0.456258
$560$	0	0
$561$	−5.54778	−0.234227
$562$	0	0
$563$	−0.183526	−0.00773469	−0.00386734	−	0.999993i	$-0.501231\pi$
$563$	−0.183526	−0.00773469	−0.00386734	+	0.999993i	$0.501231\pi$
$564$	0	0
$565$	14.2287	0.598607
$566$	0	0
$567$	6.87989	0.288928
$568$	0	0
$569$	38.3418	1.60737	0.803685	−	0.595054i	$-0.202870\pi$
$569$	38.3418	1.60737	0.803685	+	0.595054i	$0.202870\pi$
$570$	0	0
$571$	−12.1930	−0.510261	−0.255131	−	0.966907i	$-0.582118\pi$
$571$	−12.1930	−0.510261	−0.255131	+	0.966907i	$0.582118\pi$
$572$	0	0
$573$	−9.16029	−0.382677
$574$	0	0
$575$	−3.20867	−0.133811
$576$	0	0
$577$	−2.66251	−0.110842	−0.0554209	−	0.998463i	$-0.517650\pi$
$577$	−2.66251	−0.110842	−0.0554209	+	0.998463i	$0.517650\pi$
$578$	0	0
$579$	6.99983	0.290903
$580$	0	0
$581$	−7.52711	−0.312277
$582$	0	0
$583$	−21.0636	−0.872366
$584$	0	0
$585$	2.75792	0.114026
$586$	0	0
$587$	−34.4949	−1.42376	−0.711879	−	0.702302i	$-0.752155\pi$
$587$	−34.4949	−1.42376	−0.711879	+	0.702302i	$0.752155\pi$
$588$	0	0
$589$	8.28596	0.341417
$590$	0	0
$591$	−4.55544	−0.187386
$592$	0	0
$593$	−4.98863	−0.204859	−0.102429	−	0.994740i	$-0.532662\pi$
$593$	−4.98863	−0.204859	−0.102429	+	0.994740i	$0.532662\pi$
$594$	0	0
$595$	−1.82079	−0.0746452
$596$	0	0
$597$	−9.67724	−0.396063
$598$	0	0
$599$	10.5552	0.431273	0.215637	−	0.976474i	$-0.430817\pi$
$599$	10.5552	0.431273	0.215637	+	0.976474i	$0.430817\pi$
$600$	0	0
$601$	4.10507	0.167449	0.0837247	−	0.996489i	$-0.473318\pi$
$601$	4.10507	0.167449	0.0837247	+	0.996489i	$0.473318\pi$
$602$	0	0
$603$	40.8196	1.66230
$604$	0	0
$605$	27.3495	1.11192
$606$	0	0
$607$	−42.1296	−1.70999	−0.854995	−	0.518637i	$-0.826440\pi$
$607$	−42.1296	−1.70999	−0.854995	+	0.518637i	$0.826440\pi$
$608$	0	0
$609$	2.32067	0.0940381
$610$	0	0
$611$	−10.3225	−0.417605
$612$	0	0
$613$	23.7126	0.957744	0.478872	−	0.877885i	$-0.341046\pi$
$613$	23.7126	0.957744	0.478872	+	0.877885i	$0.341046\pi$
$614$	0	0
$615$	1.36480	0.0550339
$616$	0	0
$617$	−5.75432	−0.231660	−0.115830	−	0.993269i	$-0.536953\pi$
$617$	−5.75432	−0.231660	−0.115830	+	0.993269i	$0.536953\pi$
$618$	0	0
$619$	−37.5790	−1.51043	−0.755214	−	0.655478i	$-0.772467\pi$
$619$	−37.5790	−1.51043	−0.755214	+	0.655478i	$0.772467\pi$
$620$	0	0
$621$	9.09011	0.364773
$622$	0	0
$623$	2.83113	0.113427
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	10.1310	0.404591
$628$	0	0
$629$	6.62370	0.264104
$630$	0	0
$631$	−4.22735	−0.168288	−0.0841440	−	0.996454i	$-0.526816\pi$
$631$	−4.22735	−0.168288	−0.0841440	+	0.996454i	$0.526816\pi$
$632$	0	0
$633$	11.6873	0.464529
$634$	0	0
$635$	−4.30717	−0.170925
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−27.2956	−1.07980
$640$	0	0
$641$	−22.0278	−0.870046	−0.435023	−	0.900419i	$-0.643260\pi$
$641$	−22.0278	−0.870046	−0.435023	+	0.900419i	$0.643260\pi$
$642$	0	0
$643$	−9.17810	−0.361949	−0.180974	−	0.983488i	$-0.557925\pi$
$643$	−9.17810	−0.361949	−0.180974	+	0.983488i	$0.557925\pi$
$644$	0	0
$645$	5.30756	0.208985
$646$	0	0
$647$	−24.1289	−0.948604	−0.474302	−	0.880362i	$-0.657300\pi$
$647$	−24.1289	−0.948604	−0.474302	+	0.880362i	$0.657300\pi$
$648$	0	0
$649$	−60.8774	−2.38965
$650$	0	0
$651$	1.22611	0.0480550
$652$	0	0
$653$	−32.5812	−1.27500	−0.637500	−	0.770450i	$-0.720031\pi$
$653$	−32.5812	−1.27500	−0.637500	+	0.770450i	$0.720031\pi$
$654$	0	0
$655$	11.2832	0.440870
$656$	0	0
$657$	−7.50272	−0.292709
$658$	0	0
$659$	31.5643	1.22957	0.614786	−	0.788694i	$-0.289242\pi$
$659$	31.5643	1.22957	0.614786	+	0.788694i	$0.289242\pi$
$660$	0	0
$661$	47.4525	1.84569	0.922843	−	0.385175i	$-0.125859\pi$
$661$	47.4525	1.84569	0.922843	+	0.385175i	$0.125859\pi$
$662$	0	0
$663$	0.895858	0.0347923
$664$	0	0
$665$	3.32500	0.128938
$666$	0	0
$667$	−15.1342	−0.585997
$668$	0	0
$669$	7.09161	0.274177
$670$	0	0
$671$	30.4752	1.17648
$672$	0	0
$673$	−29.5081	−1.13745	−0.568726	−	0.822527i	$-0.692563\pi$
$673$	−29.5081	−1.13745	−0.568726	+	0.822527i	$0.692563\pi$
$674$	0	0
$675$	−2.83299	−0.109042
$676$	0	0
$677$	−0.303704	−0.0116723	−0.00583615	−	0.999983i	$-0.501858\pi$
$677$	−0.303704	−0.0116723	−0.00583615	+	0.999983i	$0.501858\pi$
$678$	0	0
$679$	−17.8816	−0.686232
$680$	0	0
$681$	10.8629	0.416267
$682$	0	0
$683$	−33.5165	−1.28247	−0.641236	−	0.767344i	$-0.721578\pi$
$683$	−33.5165	−1.28247	−0.641236	+	0.767344i	$0.721578\pi$
$684$	0	0
$685$	13.4680	0.514586
$686$	0	0
$687$	−12.0902	−0.461269
$688$	0	0
$689$	3.40136	0.129582
$690$	0	0
$691$	−36.6876	−1.39566	−0.697830	−	0.716263i	$-0.745851\pi$
$691$	−36.6876	−1.39566	−0.697830	+	0.716263i	$0.745851\pi$
$692$	0	0
$693$	−17.0790	−0.648776
$694$	0	0
$695$	−7.75978	−0.294345
$696$	0	0
$697$	−5.05068	−0.191308
$698$	0	0
$699$	−9.46624	−0.358046
$700$	0	0
$701$	37.3084	1.40912	0.704560	−	0.709645i	$-0.251145\pi$
$701$	37.3084	1.40912	0.704560	+	0.709645i	$0.251145\pi$
$702$	0	0
$703$	−12.0957	−0.456199
$704$	0	0
$705$	5.07884	0.191280
$706$	0	0
$707$	−1.67252	−0.0629016
$708$	0	0
$709$	−23.8552	−0.895901	−0.447950	−	0.894058i	$-0.647846\pi$
$709$	−23.8552	−0.895901	−0.447950	+	0.894058i	$0.647846\pi$
$710$	0	0
$711$	−33.4903	−1.25598
$712$	0	0
$713$	−7.99605	−0.299454
$714$	0	0
$715$	−6.19270	−0.231594
$716$	0	0
$717$	11.0474	0.412573
$718$	0	0
$719$	−30.0859	−1.12201	−0.561007	−	0.827811i	$-0.689586\pi$
$719$	−30.0859	−1.12201	−0.561007	+	0.827811i	$0.689586\pi$
$720$	0	0
$721$	8.84084	0.329250
$722$	0	0
$723$	3.89039	0.144685
$724$	0	0
$725$	4.71665	0.175172
$726$	0	0
$727$	−34.8696	−1.29324	−0.646621	−	0.762811i	$-0.723819\pi$
$727$	−34.8696	−1.29324	−0.646621	+	0.762811i	$0.723819\pi$
$728$	0	0
$729$	−14.7926	−0.547873
$730$	0	0
$731$	−19.6416	−0.726470
$732$	0	0
$733$	−47.4835	−1.75384	−0.876922	−	0.480632i	$-0.840407\pi$
$733$	−47.4835	−1.75384	−0.876922	+	0.480632i	$0.840407\pi$
$734$	0	0
$735$	0.492015	0.0181483
$736$	0	0
$737$	−91.6573	−3.37624
$738$	0	0
$739$	23.8370	0.876858	0.438429	−	0.898766i	$-0.355535\pi$
$739$	23.8370	0.876858	0.438429	+	0.898766i	$0.355535\pi$
$740$	0	0
$741$	−1.63595	−0.0600982
$742$	0	0
$743$	18.3432	0.672947	0.336473	−	0.941693i	$-0.390766\pi$
$743$	18.3432	0.672947	0.336473	+	0.941693i	$0.390766\pi$
$744$	0	0
$745$	0.838620	0.0307246
$746$	0	0
$747$	20.7592	0.759538
$748$	0	0
$749$	−4.49579	−0.164273
$750$	0	0
$751$	41.1570	1.50184	0.750920	−	0.660393i	$-0.229610\pi$
$751$	41.1570	1.50184	0.750920	+	0.660393i	$0.229610\pi$
$752$	0	0
$753$	1.12944	0.0411592
$754$	0	0
$755$	−13.4130	−0.488149
$756$	0	0
$757$	−49.4985	−1.79905	−0.899527	−	0.436864i	$-0.856089\pi$
$757$	−49.4985	−1.79905	−0.899527	+	0.436864i	$0.856089\pi$
$758$	0	0
$759$	−9.77650	−0.354864
$760$	0	0
$761$	39.2826	1.42399	0.711997	−	0.702183i	$-0.247791\pi$
$761$	39.2826	1.42399	0.711997	+	0.702183i	$0.247791\pi$
$762$	0	0
$763$	4.12574	0.149362
$764$	0	0
$765$	5.02160	0.181556
$766$	0	0
$767$	9.83051	0.354959
$768$	0	0
$769$	−42.0068	−1.51480	−0.757402	−	0.652949i	$-0.773532\pi$
$769$	−42.0068	−1.51480	−0.757402	+	0.652949i	$0.773532\pi$
$770$	0	0
$771$	3.17088	0.114196
$772$	0	0
$773$	−31.6554	−1.13857	−0.569284	−	0.822141i	$-0.692779\pi$
$773$	−31.6554	−1.13857	−0.569284	+	0.822141i	$0.692779\pi$
$774$	0	0
$775$	2.49202	0.0895158
$776$	0	0
$777$	−1.78986	−0.0642108
$778$	0	0
$779$	9.22319	0.330455
$780$	0	0
$781$	61.2901	2.19313
$782$	0	0
$783$	−13.3622	−0.477526
$784$	0	0
$785$	19.1649	0.684025
$786$	0	0
$787$	−34.0381	−1.21333	−0.606663	−	0.794959i	$-0.707492\pi$
$787$	−34.0381	−1.21333	−0.606663	+	0.794959i	$0.707492\pi$
$788$	0	0
$789$	−11.9280	−0.424647
$790$	0	0
$791$	14.2287	0.505915
$792$	0	0
$793$	−4.92116	−0.174755
$794$	0	0
$795$	−1.67352	−0.0593538
$796$	0	0
$797$	35.6224	1.26181	0.630904	−	0.775861i	$-0.282684\pi$
$797$	35.6224	1.26181	0.630904	+	0.775861i	$0.282684\pi$
$798$	0	0
$799$	−18.7952	−0.664926
$800$	0	0
$801$	−7.80803	−0.275883
$802$	0	0
$803$	16.8468	0.594510
$804$	0	0
$805$	−3.20867	−0.113091
$806$	0	0
$807$	−4.75531	−0.167395
$808$	0	0
$809$	−27.1158	−0.953342	−0.476671	−	0.879082i	$-0.658157\pi$
$809$	−27.1158	−0.953342	−0.476671	+	0.879082i	$0.658157\pi$
$810$	0	0
$811$	6.18043	0.217024	0.108512	−	0.994095i	$-0.465391\pi$
$811$	6.18043	0.217024	0.108512	+	0.994095i	$0.465391\pi$
$812$	0	0
$813$	−8.96436	−0.314394
$814$	0	0
$815$	15.2278	0.533406
$816$	0	0
$817$	35.8681	1.25486
$818$	0	0
$819$	2.75792	0.0963696
$820$	0	0
$821$	15.1462	0.528605	0.264302	−	0.964440i	$-0.414858\pi$
$821$	15.1462	0.528605	0.264302	+	0.964440i	$0.414858\pi$
$822$	0	0
$823$	−48.6208	−1.69481	−0.847407	−	0.530944i	$-0.821838\pi$
$823$	−48.6208	−1.69481	−0.847407	+	0.530944i	$0.821838\pi$
$824$	0	0
$825$	3.04690	0.106080
$826$	0	0
$827$	−26.8027	−0.932022	−0.466011	−	0.884779i	$-0.654309\pi$
$827$	−26.8027	−0.932022	−0.466011	+	0.884779i	$0.654309\pi$
$828$	0	0
$829$	−22.5306	−0.782519	−0.391260	−	0.920280i	$-0.627961\pi$
$829$	−22.5306	−0.782519	−0.391260	+	0.920280i	$0.627961\pi$
$830$	0	0
$831$	9.45619	0.328032
$832$	0	0
$833$	−1.82079	−0.0630867
$834$	0	0
$835$	−8.56375	−0.296361
$836$	0	0
$837$	−7.05985	−0.244024
$838$	0	0
$839$	−5.77242	−0.199286	−0.0996430	−	0.995023i	$-0.531770\pi$
$839$	−5.77242	−0.199286	−0.0996430	+	0.995023i	$0.531770\pi$
$840$	0	0
$841$	−6.75320	−0.232869
$842$	0	0
$843$	−0.834410	−0.0287386
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	27.3495	0.939740
$848$	0	0
$849$	−13.8332	−0.474756
$850$	0	0
$851$	11.6725	0.400129
$852$	0	0
$853$	4.28787	0.146814	0.0734070	−	0.997302i	$-0.476613\pi$
$853$	4.28787	0.146814	0.0734070	+	0.997302i	$0.476613\pi$
$854$	0	0
$855$	−9.17009	−0.313610
$856$	0	0
$857$	28.2678	0.965609	0.482805	−	0.875728i	$-0.339618\pi$
$857$	28.2678	0.965609	0.482805	+	0.875728i	$0.339618\pi$
$858$	0	0
$859$	−52.0661	−1.77647	−0.888236	−	0.459388i	$-0.848069\pi$
$859$	−52.0661	−1.77647	−0.888236	+	0.459388i	$0.848069\pi$
$860$	0	0
$861$	1.36480	0.0465121
$862$	0	0
$863$	15.7194	0.535096	0.267548	−	0.963545i	$-0.413787\pi$
$863$	15.7194	0.535096	0.267548	+	0.963545i	$0.413787\pi$
$864$	0	0
$865$	2.92493	0.0994508
$866$	0	0
$867$	−6.73309	−0.228668
$868$	0	0
$869$	75.1999	2.55098
$870$	0	0
$871$	14.8009	0.501508
$872$	0	0
$873$	49.3160	1.66909
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	6.80954	0.229942	0.114971	−	0.993369i	$-0.463322\pi$
$877$	6.80954	0.229942	0.114971	+	0.993369i	$0.463322\pi$
$878$	0	0
$879$	5.34427	0.180258
$880$	0	0
$881$	35.4647	1.19484	0.597419	−	0.801930i	$-0.296193\pi$
$881$	35.4647	1.19484	0.597419	+	0.801930i	$0.296193\pi$
$882$	0	0
$883$	47.6689	1.60419	0.802094	−	0.597198i	$-0.203719\pi$
$883$	47.6689	1.60419	0.802094	+	0.597198i	$0.203719\pi$
$884$	0	0
$885$	−4.83676	−0.162586
$886$	0	0
$887$	8.14587	0.273512	0.136756	−	0.990605i	$-0.456332\pi$
$887$	8.14587	0.273512	0.136756	+	0.990605i	$0.456332\pi$
$888$	0	0
$889$	−4.30717	−0.144458
$890$	0	0
$891$	42.6051	1.42732
$892$	0	0
$893$	34.3224	1.14856
$894$	0	0
$895$	−20.6047	−0.688739
$896$	0	0
$897$	1.57871	0.0527117
$898$	0	0
$899$	11.7540	0.392017
$900$	0	0
$901$	6.19318	0.206325
$902$	0	0
$903$	5.30756	0.176624
$904$	0	0
$905$	14.4689	0.480964
$906$	0	0
$907$	−16.4688	−0.546838	−0.273419	−	0.961895i	$-0.588155\pi$
$907$	−16.4688	−0.546838	−0.273419	+	0.961895i	$0.588155\pi$
$908$	0	0
$909$	4.61268	0.152993
$910$	0	0
$911$	57.0810	1.89118	0.945588	−	0.325366i	$-0.105487\pi$
$911$	57.0810	1.89118	0.945588	+	0.325366i	$0.105487\pi$
$912$	0	0
$913$	−46.6131	−1.54267
$914$	0	0
$915$	2.42129	0.0800453
$916$	0	0
$917$	11.2832	0.372603
$918$	0	0
$919$	24.7621	0.816825	0.408412	−	0.912797i	$-0.366083\pi$
$919$	24.7621	0.816825	0.408412	+	0.912797i	$0.366083\pi$
$920$	0	0
$921$	6.82391	0.224855
$922$	0	0
$923$	−9.89716	−0.325769
$924$	0	0
$925$	−3.63781	−0.119610
$926$	0	0
$927$	−24.3823	−0.800821
$928$	0	0
$929$	−50.7557	−1.66524	−0.832621	−	0.553843i	$-0.813161\pi$
$929$	−50.7557	−1.66524	−0.832621	+	0.553843i	$0.813161\pi$
$930$	0	0
$931$	3.32500	0.108973
$932$	0	0
$933$	15.8551	0.519072
$934$	0	0
$935$	−11.2756	−0.368752
$936$	0	0
$937$	−55.5834	−1.81583	−0.907916	−	0.419153i	$-0.862327\pi$
$937$	−55.5834	−1.81583	−0.907916	+	0.419153i	$0.862327\pi$
$938$	0	0
$939$	−3.36656	−0.109864
$940$	0	0
$941$	−35.3906	−1.15370	−0.576850	−	0.816850i	$-0.695718\pi$
$941$	−35.3906	−1.15370	−0.576850	+	0.816850i	$0.695718\pi$
$942$	0	0
$943$	−8.90049	−0.289840
$944$	0	0
$945$	−2.83299	−0.0921570
$946$	0	0
$947$	2.31622	0.0752670	0.0376335	−	0.999292i	$-0.488018\pi$
$947$	2.31622	0.0752670	0.0376335	+	0.999292i	$0.488018\pi$
$948$	0	0
$949$	−2.72043	−0.0883088
$950$	0	0
$951$	0.509549	0.0165233
$952$	0	0
$953$	52.5470	1.70217	0.851083	−	0.525031i	$-0.175946\pi$
$953$	52.5470	1.70217	0.851083	+	0.525031i	$0.175946\pi$
$954$	0	0
$955$	−18.6179	−0.602461
$956$	0	0
$957$	14.3712	0.464554
$958$	0	0
$959$	13.4680	0.434905
$960$	0	0
$961$	−24.7899	−0.799673
$962$	0	0
$963$	12.3990	0.399553
$964$	0	0
$965$	14.2269	0.457979
$966$	0	0
$967$	−31.0860	−0.999658	−0.499829	−	0.866124i	$-0.666604\pi$
$967$	−31.0860	−0.999658	−0.499829	+	0.866124i	$0.666604\pi$
$968$	0	0
$969$	−2.97873	−0.0956906
$970$	0	0
$971$	−58.8258	−1.88781	−0.943904	−	0.330219i	$-0.892877\pi$
$971$	−58.8258	−1.88781	−0.943904	+	0.330219i	$0.892877\pi$
$972$	0	0
$973$	−7.75978	−0.248767
$974$	0	0
$975$	−0.492015	−0.0157571
$976$	0	0
$977$	21.2578	0.680098	0.340049	−	0.940408i	$-0.389556\pi$
$977$	21.2578	0.680098	0.340049	+	0.940408i	$0.389556\pi$
$978$	0	0
$979$	17.5323	0.560335
$980$	0	0
$981$	−11.3785	−0.363287
$982$	0	0
$983$	31.9125	1.01785	0.508925	−	0.860811i	$-0.330043\pi$
$983$	31.9125	1.01785	0.508925	+	0.860811i	$0.330043\pi$
$984$	0	0
$985$	−9.25872	−0.295008
$986$	0	0
$987$	5.07884	0.161661
$988$	0	0
$989$	−34.6131	−1.10063
$990$	0	0
$991$	25.2144	0.800961	0.400480	−	0.916305i	$-0.368843\pi$
$991$	25.2144	0.800961	0.400480	+	0.916305i	$0.368843\pi$
$992$	0	0
$993$	−15.7615	−0.500178
$994$	0	0
$995$	−19.6686	−0.623535
$996$	0	0
$997$	−47.5048	−1.50449	−0.752247	−	0.658882i	$-0.771030\pi$
$997$	−47.5048	−1.50449	−0.752247	+	0.658882i	$0.771030\pi$
$998$	0	0
$999$	10.3059	0.326063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.z.1.3	✓	5
4.3	odd	2		7280.2.a.cd.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.z.1.3	✓	5	1.1	even	1	trivial
7280.2.a.cd.1.3		5	4.3	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:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q+0.492015 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.75792 q^{9} +O(q^{10})\)	\(q+0.492015 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.75792 q^{9} +6.19270 q^{11} -1.00000 q^{13} +0.492015 q^{15} -1.82079 q^{17} +3.32500 q^{19} +0.492015 q^{21} -3.20867 q^{23} +1.00000 q^{25} -2.83299 q^{27} +4.71665 q^{29} +2.49202 q^{31} +3.04690 q^{33} +1.00000 q^{35} -3.63781 q^{37} -0.492015 q^{39} +2.77389 q^{41} +10.7874 q^{43} -2.75792 q^{45} +10.3225 q^{47} +1.00000 q^{49} -0.895858 q^{51} -3.40136 q^{53} +6.19270 q^{55} +1.63595 q^{57} -9.83051 q^{59} +4.92116 q^{61} -2.75792 q^{63} -1.00000 q^{65} -14.8009 q^{67} -1.57871 q^{69} +9.89716 q^{71} +2.72043 q^{73} +0.492015 q^{75} +6.19270 q^{77} +12.1433 q^{79} +6.87989 q^{81} -7.52711 q^{83} -1.82079 q^{85} +2.32067 q^{87} +2.83113 q^{89} -1.00000 q^{91} +1.22611 q^{93} +3.32500 q^{95} -17.8816 q^{97} -17.0790 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9	\( 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9} + 7 q^{11} - 5 q^{13} + 3 q^{17} + 3 q^{19} + 3 q^{23} + 5 q^{25} - 3 q^{27} + 7 q^{29} + 10 q^{31} + 17 q^{33} + 5 q^{35} + 10 q^{37} + 4 q^{43} + 5 q^{45} - 3 q^{47} + 5 q^{49} + 26 q^{53} + 7 q^{55} - 20 q^{57} + 3 q^{59} + 13 q^{61} + 5 q^{63} - 5 q^{65} + 12 q^{67} + 23 q^{69} + 8 q^{71} + q^{73} + 7 q^{77} - 6 q^{79} + 25 q^{81} - 8 q^{83} + 3 q^{85} - 43 q^{87} + 3 q^{89} - 5 q^{91} + 20 q^{93} + 3 q^{95} - 15 q^{97} + 16 q^{99}+O(q^{100}) \) 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9 + 7 * q^11 - 5 * q^13 + 3 * q^17 + 3 * q^19 + 3 * q^23 + 5 * q^25 - 3 * q^27 + 7 * q^29 + 10 * q^31 + 17 * q^33 + 5 * q^35 + 10 * q^37 + 4 * q^43 + 5 * q^45 - 3 * q^47 + 5 * q^49 + 26 * q^53 + 7 * q^55 - 20 * q^57 + 3 * q^59 + 13 * q^61 + 5 * q^63 - 5 * q^65 + 12 * q^67 + 23 * q^69 + 8 * q^71 + q^73 + 7 * q^77 - 6 * q^79 + 25 * q^81 - 8 * q^83 + 3 * q^85 - 43 * q^87 + 3 * q^89 - 5 * q^91 + 20 * q^93 + 3 * q^95 - 15 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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