LMFDB - Embedded newform 6724.2.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6724,2,Mod(1,6724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6724, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6724 = 2^{2} \cdot 41^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6724.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:	$53.6914103191$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.25808.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} - 6x + 9 $ x^4 - 10x^2 - 6x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 164)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.31526$ of defining polynomial
Character	$\chi$	$=$	6724.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0950939 q^{3} +1.17025 q^{5} -3.14501 q^{7} -2.99096 q^{9} +O(q^{10})$	$q-0.0950939 q^{3} +1.17025 q^{5} -3.14501 q^{7} -2.99096 q^{9} +1.67570 q^{11} -6.63052 q^{13} -0.111283 q^{15} -5.16120 q^{17} +4.72562 q^{19} +0.299072 q^{21} -8.82071 q^{23} -3.63052 q^{25} +0.569704 q^{27} +1.80981 q^{29} -1.65951 q^{31} -0.159349 q^{33} -3.68044 q^{35} -1.99096 q^{37} +0.630522 q^{39} +1.46932 q^{43} -3.50016 q^{45} +8.53543 q^{47} +2.89112 q^{49} +0.490799 q^{51} +9.35139 q^{53} +1.96098 q^{55} -0.449377 q^{57} +8.82071 q^{59} +12.6305 q^{61} +9.40660 q^{63} -7.75935 q^{65} -9.67570 q^{67} +0.838796 q^{69} +0.776081 q^{71} +8.33145 q^{73} +0.345241 q^{75} -5.27009 q^{77} -0.915804 q^{79} +8.91870 q^{81} -10.0998 q^{83} -6.03988 q^{85} -0.172102 q^{87} +6.44033 q^{89} +20.8531 q^{91} +0.157809 q^{93} +5.53014 q^{95} +8.32241 q^{97} -5.01193 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 4 q^{5} + 12 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 4 * q^5 + 12 * q^9	$ 4 q - 2 q^{3} + 4 q^{5} + 12 q^{9} - 4 q^{11} + 10 q^{15} + 4 q^{17} - 6 q^{19} - 12 q^{23} + 12 q^{25} + 10 q^{27} + 4 q^{29} - 8 q^{31} - 20 q^{33} + 26 q^{35} + 16 q^{37} - 24 q^{39} + 4 q^{43} + 4 q^{45} + 6 q^{47} + 16 q^{49} - 4 q^{51} + 16 q^{53} + 2 q^{55} + 4 q^{57} + 12 q^{59} + 24 q^{61} + 10 q^{63} - 4 q^{65} - 28 q^{67} + 28 q^{69} + 2 q^{71} + 8 q^{73} - 30 q^{75} + 8 q^{77} + 18 q^{79} + 28 q^{81} - 12 q^{83} - 32 q^{85} + 44 q^{87} - 4 q^{89} + 36 q^{91} + 28 q^{93} - 14 q^{95} - 16 q^{97} - 58 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 4 * q^5 + 12 * q^9 - 4 * q^11 + 10 * q^15 + 4 * q^17 - 6 * q^19 - 12 * q^23 + 12 * q^25 + 10 * q^27 + 4 * q^29 - 8 * q^31 - 20 * q^33 + 26 * q^35 + 16 * q^37 - 24 * q^39 + 4 * q^43 + 4 * q^45 + 6 * q^47 + 16 * q^49 - 4 * q^51 + 16 * q^53 + 2 * q^55 + 4 * q^57 + 12 * q^59 + 24 * q^61 + 10 * q^63 - 4 * q^65 - 28 * q^67 + 28 * q^69 + 2 * q^71 + 8 * q^73 - 30 * q^75 + 8 * q^77 + 18 * q^79 + 28 * q^81 - 12 * q^83 - 32 * q^85 + 44 * q^87 - 4 * q^89 + 36 * q^91 + 28 * q^93 - 14 * q^95 - 16 * q^97 - 58 * 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
$3$	−0.0950939	−0.0549025	−0.0274513	−	0.999623i	$-0.508739\pi$
$3$	−0.0950939	−0.0549025	−0.0274513	+	0.999623i	$0.508739\pi$
$4$	0	0
$5$	1.17025	0.523350	0.261675	−	0.965156i	$-0.415725\pi$
$5$	1.17025	0.523350	0.261675	+	0.965156i	$0.415725\pi$
$6$	0	0
$7$	−3.14501	−1.18870	−0.594352	−	0.804205i	$-0.702591\pi$
$7$	−3.14501	−1.18870	−0.594352	+	0.804205i	$0.702591\pi$
$8$	0	0
$9$	−2.99096	−0.996986
$10$	0	0
$11$	1.67570	0.505241	0.252621	−	0.967565i	$-0.418708\pi$
$11$	1.67570	0.505241	0.252621	+	0.967565i	$0.418708\pi$
$12$	0	0
$13$	−6.63052	−1.83898	−0.919488	−	0.393118i	$-0.871396\pi$
$13$	−6.63052	−1.83898	−0.919488	+	0.393118i	$0.871396\pi$
$14$	0	0
$15$	−0.111283	−0.0287332
$16$	0	0
$17$	−5.16120	−1.25178	−0.625888	−	0.779913i	$-0.715263\pi$
$17$	−5.16120	−1.25178	−0.625888	+	0.779913i	$0.715263\pi$
$18$	0	0
$19$	4.72562	1.08413	0.542065	−	0.840336i	$-0.317643\pi$
$19$	4.72562	1.08413	0.542065	+	0.840336i	$0.317643\pi$
$20$	0	0
$21$	0.299072	0.0652628
$22$	0	0
$23$	−8.82071	−1.83925	−0.919623	−	0.392803i	$-0.871505\pi$
$23$	−8.82071	−1.83925	−0.919623	+	0.392803i	$0.871505\pi$
$24$	0	0
$25$	−3.63052	−0.726104
$26$	0	0
$27$	0.569704	0.109640
$28$	0	0
$29$	1.80981	0.336074	0.168037	−	0.985781i	$-0.446257\pi$
$29$	1.80981	0.336074	0.168037	+	0.985781i	$0.446257\pi$
$30$	0	0
$31$	−1.65951	−0.298056	−0.149028	−	0.988833i	$-0.547614\pi$
$31$	−1.65951	−0.298056	−0.149028	+	0.988833i	$0.547614\pi$
$32$	0	0
$33$	−0.159349	−0.0277390
$34$	0	0
$35$	−3.68044	−0.622108
$36$	0	0
$37$	−1.99096	−0.327311	−0.163656	−	0.986518i	$-0.552329\pi$
$37$	−1.99096	−0.327311	−0.163656	+	0.986518i	$0.552329\pi$
$38$	0	0
$39$	0.630522	0.100964
$40$	0	0
$41$	0	0
$41$	0	0
$42$	0	0
$43$	1.46932	0.224069	0.112034	−	0.993704i	$-0.464263\pi$
$43$	1.46932	0.224069	0.112034	+	0.993704i	$0.464263\pi$
$44$	0	0
$45$	−3.50016	−0.521773
$46$	0	0
$47$	8.53543	1.24502	0.622510	−	0.782612i	$-0.286113\pi$
$47$	8.53543	1.24502	0.622510	+	0.782612i	$0.286113\pi$
$48$	0	0
$49$	2.89112	0.413017
$50$	0	0
$51$	0.490799	0.0687256
$52$	0	0
$53$	9.35139	1.28451	0.642256	−	0.766490i	$-0.277999\pi$
$53$	9.35139	1.28451	0.642256	+	0.766490i	$0.277999\pi$
$54$	0	0
$55$	1.96098	0.264418
$56$	0	0
$57$	−0.449377	−0.0595215
$58$	0	0
$59$	8.82071	1.14836	0.574179	−	0.818730i	$-0.305321\pi$
$59$	8.82071	1.14836	0.574179	+	0.818730i	$0.305321\pi$
$60$	0	0
$61$	12.6305	1.61717	0.808586	−	0.588378i	$-0.200233\pi$
$61$	12.6305	1.61717	0.808586	+	0.588378i	$0.200233\pi$
$62$	0	0
$63$	9.40660	1.18512
$64$	0	0
$65$	−7.75935	−0.962429
$66$	0	0
$67$	−9.67570	−1.18207	−0.591037	−	0.806644i	$-0.701281\pi$
$67$	−9.67570	−1.18207	−0.591037	+	0.806644i	$0.701281\pi$
$68$	0	0
$69$	0.838796	0.100979
$70$	0	0
$71$	0.776081	0.0921039	0.0460519	−	0.998939i	$-0.485336\pi$
$71$	0.776081	0.0921039	0.0460519	+	0.998939i	$0.485336\pi$
$72$	0	0
$73$	8.33145	0.975123	0.487561	−	0.873089i	$-0.337887\pi$
$73$	8.33145	0.975123	0.487561	+	0.873089i	$0.337887\pi$
$74$	0	0
$75$	0.345241	0.0398650
$76$	0	0
$77$	−5.27009	−0.600582
$78$	0	0
$79$	−0.915804	−0.103036	−0.0515180	−	0.998672i	$-0.516406\pi$
$79$	−0.915804	−0.103036	−0.0515180	+	0.998672i	$0.516406\pi$
$80$	0	0
$81$	8.91870	0.990966
$82$	0	0
$83$	−10.0998	−1.10860	−0.554301	−	0.832316i	$-0.687014\pi$
$83$	−10.0998	−1.10860	−0.554301	+	0.832316i	$0.687014\pi$
$84$	0	0
$85$	−6.03988	−0.655117
$86$	0	0
$87$	−0.172102	−0.0184513
$88$	0	0
$89$	6.44033	0.682674	0.341337	−	0.939941i	$-0.389120\pi$
$89$	6.44033	0.682674	0.341337	+	0.939941i	$0.389120\pi$
$90$	0	0
$91$	20.8531	2.18600
$92$	0	0
$93$	0.157809	0.0163640
$94$	0	0
$95$	5.53014	0.567380
$96$	0	0
$97$	8.32241	0.845012	0.422506	−	0.906360i	$-0.361150\pi$
$97$	8.32241	0.845012	0.422506	+	0.906360i	$0.361150\pi$
$98$	0	0
$99$	−5.01193	−0.503718
$100$	0	0
$101$	−9.79173	−0.974313	−0.487157	−	0.873315i	$-0.661966\pi$
$101$	−9.79173	−0.974313	−0.487157	+	0.873315i	$0.661966\pi$
$102$	0	0
$103$	−0.648608	−0.0639093	−0.0319546	−	0.999489i	$-0.510173\pi$
$103$	−0.648608	−0.0639093	−0.0319546	+	0.999489i	$0.510173\pi$
$104$	0	0
$105$	0.349988	0.0341553
$106$	0	0
$107$	16.8026	1.62437	0.812186	−	0.583399i	$-0.198278\pi$
$107$	16.8026	1.62437	0.812186	+	0.583399i	$0.198278\pi$
$108$	0	0
$109$	−3.84969	−0.368734	−0.184367	−	0.982857i	$-0.559023\pi$
$109$	−3.84969	−0.368734	−0.184367	+	0.982857i	$0.559023\pi$
$110$	0	0
$111$	0.189328	0.0179702
$112$	0	0
$113$	−7.65046	−0.719695	−0.359848	−	0.933011i	$-0.617171\pi$
$113$	−7.65046	−0.719695	−0.359848	+	0.933011i	$0.617171\pi$
$114$	0	0
$115$	−10.3224	−0.962569
$116$	0	0
$117$	19.8316	1.83343
$118$	0	0
$119$	16.2321	1.48799
$120$	0	0
$121$	−8.19204	−0.744731
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.0998	−0.903357
$126$	0	0
$127$	14.9206	1.32398	0.661992	−	0.749511i	$-0.269711\pi$
$127$	14.9206	1.32398	0.661992	+	0.749511i	$0.269711\pi$
$128$	0	0
$129$	−0.139723	−0.0123019
$130$	0	0
$131$	4.19019	0.366098	0.183049	−	0.983104i	$-0.441403\pi$
$131$	4.19019	0.366098	0.183049	+	0.983104i	$0.441403\pi$
$132$	0	0
$133$	−14.8621	−1.28871
$134$	0	0
$135$	0.666694	0.0573799
$136$	0	0
$137$	−4.09984	−0.350273	−0.175137	−	0.984544i	$-0.556037\pi$
$137$	−4.09984	−0.350273	−0.175137	+	0.984544i	$0.556037\pi$
$138$	0	0
$139$	−11.9819	−1.01629	−0.508146	−	0.861271i	$-0.669669\pi$
$139$	−11.9819	−1.01629	−0.508146	+	0.861271i	$0.669669\pi$
$140$	0	0
$141$	−0.811668	−0.0683547
$142$	0	0
$143$	−11.1107	−0.929127
$144$	0	0
$145$	2.11793	0.175884
$146$	0	0
$147$	−0.274928	−0.0226756
$148$	0	0
$149$	1.80981	0.148266	0.0741328	−	0.997248i	$-0.476381\pi$
$149$	1.80981	0.148266	0.0741328	+	0.997248i	$0.476381\pi$
$150$	0	0
$151$	21.3561	1.73794	0.868969	−	0.494867i	$-0.164783\pi$
$151$	21.3561	1.73794	0.868969	+	0.494867i	$0.164783\pi$
$152$	0	0
$153$	15.4369	1.24800
$154$	0	0
$155$	−1.94203	−0.155988
$156$	0	0
$157$	−13.1107	−1.04635	−0.523175	−	0.852225i	$-0.675253\pi$
$157$	−13.1107	−1.04635	−0.523175	+	0.852225i	$0.675253\pi$
$158$	0	0
$159$	−0.889261	−0.0705230
$160$	0	0
$161$	27.7413	2.18632
$162$	0	0
$163$	10.7808	0.844420	0.422210	−	0.906498i	$-0.361255\pi$
$163$	10.7808	0.844420	0.422210	+	0.906498i	$0.361255\pi$
$164$	0	0
$165$	−0.186477	−0.0145172
$166$	0	0
$167$	−13.6757	−1.05826	−0.529129	−	0.848542i	$-0.677481\pi$
$167$	−13.6757	−1.05826	−0.529129	+	0.848542i	$0.677481\pi$
$168$	0	0
$169$	30.9638	2.38183
$170$	0	0
$171$	−14.1341	−1.08086
$172$	0	0
$173$	10.5801	0.804387	0.402193	−	0.915555i	$-0.368248\pi$
$173$	10.5801	0.804387	0.402193	+	0.915555i	$0.368248\pi$
$174$	0	0
$175$	11.4180	0.863123
$176$	0	0
$177$	−0.838796	−0.0630478
$178$	0	0
$179$	1.23536	0.0923352	0.0461676	−	0.998934i	$-0.485299\pi$
$179$	1.23536	0.0923352	0.0461676	+	0.998934i	$0.485299\pi$
$180$	0	0
$181$	3.90965	0.290602	0.145301	−	0.989387i	$-0.453585\pi$
$181$	3.90965	0.290602	0.145301	+	0.989387i	$0.453585\pi$
$182$	0	0
$183$	−1.20109	−0.0887868
$184$	0	0
$185$	−2.32991	−0.171298
$186$	0	0
$187$	−8.64861	−0.632449
$188$	0	0
$189$	−1.79173	−0.130329
$190$	0	0
$191$	7.31712	0.529448	0.264724	−	0.964324i	$-0.414719\pi$
$191$	7.31712	0.529448	0.264724	+	0.964324i	$0.414719\pi$
$192$	0	0
$193$	13.3009	0.957422	0.478711	−	0.877973i	$-0.341104\pi$
$193$	13.3009	0.957422	0.478711	+	0.877973i	$0.341104\pi$
$194$	0	0
$195$	0.737867	0.0528397
$196$	0	0
$197$	−15.5692	−1.10926	−0.554628	−	0.832098i	$-0.687140\pi$
$197$	−15.5692	−1.10926	−0.554628	+	0.832098i	$0.687140\pi$
$198$	0	0
$199$	27.8972	1.97758	0.988789	−	0.149319i	$-0.0477080\pi$
$199$	27.8972	1.97758	0.988789	+	0.149319i	$0.0477080\pi$
$200$	0	0
$201$	0.920100	0.0648989
$202$	0	0
$203$	−5.69189	−0.399492
$204$	0	0
$205$	0	0
$206$	0	0
$207$	26.3824	1.83370
$208$	0	0
$209$	7.91870	0.547748
$210$	0	0
$211$	−18.1484	−1.24939	−0.624694	−	0.780870i	$-0.714776\pi$
$211$	−18.1484	−1.24939	−0.624694	+	0.780870i	$0.714776\pi$
$212$	0	0
$213$	−0.0738006	−0.00505673
$214$	0	0
$215$	1.71947	0.117267
$216$	0	0
$217$	5.21917	0.354300
$218$	0	0
$219$	−0.792270	−0.0535367
$220$	0	0
$221$	34.2215	2.30199
$222$	0	0
$223$	−6.55826	−0.439174	−0.219587	−	0.975593i	$-0.570471\pi$
$223$	−6.55826	−0.439174	−0.219587	+	0.975593i	$0.570471\pi$
$224$	0	0
$225$	10.8587	0.723916
$226$	0	0
$227$	−23.4884	−1.55898	−0.779489	−	0.626416i	$-0.784521\pi$
$227$	−23.4884	−1.55898	−0.779489	+	0.626416i	$0.784521\pi$
$228$	0	0
$229$	−12.1503	−0.802915	−0.401457	−	0.915878i	$-0.631496\pi$
$229$	−12.1503	−0.802915	−0.401457	+	0.915878i	$0.631496\pi$
$230$	0	0
$231$	0.501153	0.0329735
$232$	0	0
$233$	21.8422	1.43093	0.715465	−	0.698649i	$-0.246215\pi$
$233$	21.8422	1.43093	0.715465	+	0.698649i	$0.246215\pi$
$234$	0	0
$235$	9.98856	0.651582
$236$	0	0
$237$	0.0870874	0.00565694
$238$	0	0
$239$	−21.5968	−1.39698	−0.698490	−	0.715620i	$-0.746144\pi$
$239$	−21.5968	−1.39698	−0.698490	+	0.715620i	$0.746144\pi$
$240$	0	0
$241$	12.6305	0.813603	0.406802	−	0.913516i	$-0.366644\pi$
$241$	12.6305	0.813603	0.406802	+	0.913516i	$0.366644\pi$
$242$	0	0
$243$	−2.55723	−0.164046
$244$	0	0
$245$	3.38332	0.216152
$246$	0	0
$247$	−31.3333	−1.99369
$248$	0	0
$249$	0.960434	0.0608650
$250$	0	0
$251$	20.7702	1.31101	0.655503	−	0.755192i	$-0.272457\pi$
$251$	20.7702	1.31101	0.655503	+	0.755192i	$0.272457\pi$
$252$	0	0
$253$	−14.7808	−0.929263
$254$	0	0
$255$	0.574356	0.0359676
$256$	0	0
$257$	−14.8207	−0.924491	−0.462245	−	0.886752i	$-0.652956\pi$
$257$	−14.8207	−0.924491	−0.462245	+	0.886752i	$0.652956\pi$
$258$	0	0
$259$	6.26159	0.389076
$260$	0	0
$261$	−5.41307	−0.335061
$262$	0	0
$263$	−18.4175	−1.13567	−0.567836	−	0.823142i	$-0.692219\pi$
$263$	−18.4175	−1.13567	−0.567836	+	0.823142i	$0.692219\pi$
$264$	0	0
$265$	10.9434	0.672250
$266$	0	0
$267$	−0.612437	−0.0374805
$268$	0	0
$269$	−25.8916	−1.57864	−0.789318	−	0.613984i	$-0.789566\pi$
$269$	−25.8916	−1.57864	−0.789318	+	0.613984i	$0.789566\pi$
$270$	0	0
$271$	−17.5017	−1.06315	−0.531576	−	0.847010i	$-0.678400\pi$
$271$	−17.5017	−1.06315	−0.531576	+	0.847010i	$0.678400\pi$
$272$	0	0
$273$	−1.98300	−0.120017
$274$	0	0
$275$	−6.08365	−0.366858
$276$	0	0
$277$	8.77179	0.527045	0.263523	−	0.964653i	$-0.415116\pi$
$277$	8.77179	0.527045	0.263523	+	0.964653i	$0.415116\pi$
$278$	0	0
$279$	4.96351	0.297158
$280$	0	0
$281$	−16.2215	−0.967692	−0.483846	−	0.875153i	$-0.660761\pi$
$281$	−16.2215	−0.967692	−0.483846	+	0.875153i	$0.660761\pi$
$282$	0	0
$283$	−11.9819	−0.712251	−0.356125	−	0.934438i	$-0.615902\pi$
$283$	−11.9819	−0.712251	−0.356125	+	0.934438i	$0.615902\pi$
$284$	0	0
$285$	−0.525883	−0.0311506
$286$	0	0
$287$	0	0
$288$	0	0
$289$	9.63803	0.566943
$290$	0	0
$291$	−0.791411	−0.0463933
$292$	0	0
$293$	11.2516	0.657323	0.328661	−	0.944448i	$-0.393403\pi$
$293$	11.2516	0.657323	0.328661	+	0.944448i	$0.393403\pi$
$294$	0	0
$295$	10.3224	0.600994
$296$	0	0
$297$	0.954650	0.0553944
$298$	0	0
$299$	58.4859	3.38233
$300$	0	0
$301$	−4.62103	−0.266352
$302$	0	0
$303$	0.931134	0.0534922
$304$	0	0
$305$	14.7808	0.846348
$306$	0	0
$307$	8.61244	0.491538	0.245769	−	0.969328i	$-0.420960\pi$
$307$	8.61244	0.491538	0.245769	+	0.969328i	$0.420960\pi$
$308$	0	0
$309$	0.0616787	0.00350878
$310$	0	0
$311$	−31.7575	−1.80080	−0.900400	−	0.435063i	$-0.856726\pi$
$311$	−31.7575	−1.80080	−0.900400	+	0.435063i	$0.856726\pi$
$312$	0	0
$313$	−2.21777	−0.125356	−0.0626778	−	0.998034i	$-0.519964\pi$
$313$	−2.21777	−0.125356	−0.0626778	+	0.998034i	$0.519964\pi$
$314$	0	0
$315$	11.0080	0.620233
$316$	0	0
$317$	18.3947	1.03315	0.516574	−	0.856243i	$-0.327207\pi$
$317$	18.3947	1.03315	0.516574	+	0.856243i	$0.327207\pi$
$318$	0	0
$319$	3.03269	0.169798
$320$	0	0
$321$	−1.59783	−0.0891820
$322$	0	0
$323$	−24.3899	−1.35709
$324$	0	0
$325$	24.0723	1.33529
$326$	0	0
$327$	0.366083	0.0202444
$328$	0	0
$329$	−26.8440	−1.47996
$330$	0	0
$331$	−10.1160	−0.556027	−0.278014	−	0.960577i	$-0.589676\pi$
$331$	−10.1160	−0.556027	−0.278014	+	0.960577i	$0.589676\pi$
$332$	0	0
$333$	5.95487	0.326325
$334$	0	0
$335$	−11.3230	−0.618639
$336$	0	0
$337$	7.81135	0.425511	0.212756	−	0.977105i	$-0.431756\pi$
$337$	7.81135	0.425511	0.212756	+	0.977105i	$0.431756\pi$
$338$	0	0
$339$	0.727513	0.0395131
$340$	0	0
$341$	−2.78083	−0.150590
$342$	0	0
$343$	12.9225	0.697749
$344$	0	0
$345$	0.981598	0.0528475
$346$	0	0
$347$	15.6281	0.838959	0.419480	−	0.907765i	$-0.362213\pi$
$347$	15.6281	0.838959	0.419480	+	0.907765i	$0.362213\pi$
$348$	0	0
$349$	7.49265	0.401073	0.200536	−	0.979686i	$-0.435732\pi$
$349$	7.49265	0.401073	0.200536	+	0.979686i	$0.435732\pi$
$350$	0	0
$351$	−3.77743	−0.201624
$352$	0	0
$353$	22.4313	1.19390	0.596949	−	0.802279i	$-0.296380\pi$
$353$	22.4313	1.19390	0.596949	+	0.802279i	$0.296380\pi$
$354$	0	0
$355$	0.908206	0.0482026
$356$	0	0
$357$	−1.54357	−0.0816944
$358$	0	0
$359$	−15.6196	−0.824372	−0.412186	−	0.911100i	$-0.635235\pi$
$359$	−15.6196	−0.824372	−0.412186	+	0.911100i	$0.635235\pi$
$360$	0	0
$361$	3.33145	0.175340
$362$	0	0
$363$	0.779014	0.0408876
$364$	0	0
$365$	9.74985	0.510331
$366$	0	0
$367$	−3.82790	−0.199815	−0.0999073	−	0.994997i	$-0.531855\pi$
$367$	−3.82790	−0.199815	−0.0999073	+	0.994997i	$0.531855\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−29.4103	−1.52690
$372$	0	0
$373$	−18.1997	−0.942344	−0.471172	−	0.882041i	$-0.656169\pi$
$373$	−18.1997	−0.942344	−0.471172	+	0.882041i	$0.656169\pi$
$374$	0	0
$375$	0.960434	0.0495966
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−14.7627	−0.758311	−0.379156	−	0.925333i	$-0.623785\pi$
$379$	−14.7627	−0.758311	−0.379156	+	0.925333i	$0.623785\pi$
$380$	0	0
$381$	−1.41885	−0.0726901
$382$	0	0
$383$	−4.75406	−0.242921	−0.121460	−	0.992596i	$-0.538758\pi$
$383$	−4.75406	−0.242921	−0.121460	+	0.992596i	$0.538758\pi$
$384$	0	0
$385$	−6.16730	−0.314315
$386$	0	0
$387$	−4.39467	−0.223394
$388$	0	0
$389$	14.3804	0.729114	0.364557	−	0.931181i	$-0.381220\pi$
$389$	14.3804	0.729114	0.364557	+	0.931181i	$0.381220\pi$
$390$	0	0
$391$	45.5255	2.30232
$392$	0	0
$393$	−0.398461	−0.0200997
$394$	0	0
$395$	−1.07172	−0.0539239
$396$	0	0
$397$	10.9311	0.548618	0.274309	−	0.961642i	$-0.411551\pi$
$397$	10.9311	0.548618	0.274309	+	0.961642i	$0.411551\pi$
$398$	0	0
$399$	1.41330	0.0707534
$400$	0	0
$401$	30.4132	1.51876	0.759382	−	0.650646i	$-0.225502\pi$
$401$	30.4132	1.51876	0.759382	+	0.650646i	$0.225502\pi$
$402$	0	0
$403$	11.0034	0.548118
$404$	0	0
$405$	10.4371	0.518622
$406$	0	0
$407$	−3.33624	−0.165371
$408$	0	0
$409$	13.3100	0.658136	0.329068	−	0.944306i	$-0.393266\pi$
$409$	13.3100	0.658136	0.329068	+	0.944306i	$0.393266\pi$
$410$	0	0
$411$	0.389870	0.0192309
$412$	0	0
$413$	−27.7413	−1.36506
$414$	0	0
$415$	−11.8193	−0.580187
$416$	0	0
$417$	1.13941	0.0557970
$418$	0	0
$419$	15.7917	0.771476	0.385738	−	0.922608i	$-0.373947\pi$
$419$	15.7917	0.771476	0.385738	+	0.922608i	$0.373947\pi$
$420$	0	0
$421$	34.4317	1.67810	0.839050	−	0.544054i	$-0.183111\pi$
$421$	34.4317	1.67810	0.839050	+	0.544054i	$0.183111\pi$
$422$	0	0
$423$	−25.5291	−1.24127
$424$	0	0
$425$	18.7379	0.908920
$426$	0	0
$427$	−39.7232	−1.92234
$428$	0	0
$429$	1.05656	0.0510114
$430$	0	0
$431$	5.69189	0.274168	0.137084	−	0.990559i	$-0.456227\pi$
$431$	5.69189	0.274168	0.137084	+	0.990559i	$0.456227\pi$
$432$	0	0
$433$	−13.0034	−0.624903	−0.312452	−	0.949934i	$-0.601150\pi$
$433$	−13.0034	−0.624903	−0.312452	+	0.949934i	$0.601150\pi$
$434$	0	0
$435$	−0.201402	−0.00965649
$436$	0	0
$437$	−41.6833	−1.99398
$438$	0	0
$439$	−9.59593	−0.457989	−0.228994	−	0.973428i	$-0.573544\pi$
$439$	−9.59593	−0.457989	−0.228994	+	0.973428i	$0.573544\pi$
$440$	0	0
$441$	−8.64720	−0.411772
$442$	0	0
$443$	−2.47072	−0.117388	−0.0586938	−	0.998276i	$-0.518694\pi$
$443$	−2.47072	−0.117388	−0.0586938	+	0.998276i	$0.518694\pi$
$444$	0	0
$445$	7.53678	0.357278
$446$	0	0
$447$	−0.172102	−0.00814015
$448$	0	0
$449$	35.2324	1.66272	0.831359	−	0.555735i	$-0.187563\pi$
$449$	35.2324	1.66272	0.831359	+	0.555735i	$0.187563\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−2.03084	−0.0954172
$454$	0	0
$455$	24.4033	1.14404
$456$	0	0
$457$	37.0241	1.73191	0.865957	−	0.500118i	$-0.166710\pi$
$457$	37.0241	1.73191	0.865957	+	0.500118i	$0.166710\pi$
$458$	0	0
$459$	−2.94036	−0.137244
$460$	0	0
$461$	−27.6324	−1.28697	−0.643484	−	0.765460i	$-0.722512\pi$
$461$	−27.6324	−1.28697	−0.643484	+	0.765460i	$0.722512\pi$
$462$	0	0
$463$	−23.1877	−1.07763	−0.538813	−	0.842425i	$-0.681127\pi$
$463$	−23.1877	−1.07763	−0.538813	+	0.842425i	$0.681127\pi$
$464$	0	0
$465$	0.184675	0.00856412
$466$	0	0
$467$	−27.5235	−1.27364	−0.636818	−	0.771014i	$-0.719750\pi$
$467$	−27.5235	−1.27364	−0.636818	+	0.771014i	$0.719750\pi$
$468$	0	0
$469$	30.4302	1.40514
$470$	0	0
$471$	1.24675	0.0574473
$472$	0	0
$473$	2.46213	0.113209
$474$	0	0
$475$	−17.1565	−0.787192
$476$	0	0
$477$	−27.9696	−1.28064
$478$	0	0
$479$	−11.1088	−0.507576	−0.253788	−	0.967260i	$-0.581677\pi$
$479$	−11.1088	−0.507576	−0.253788	+	0.967260i	$0.581677\pi$
$480$	0	0
$481$	13.2011	0.601918
$482$	0	0
$483$	−2.63803	−0.120034
$484$	0	0
$485$	9.73927	0.442238
$486$	0	0
$487$	−31.3933	−1.42256	−0.711282	−	0.702906i	$-0.751885\pi$
$487$	−31.3933	−1.42256	−0.711282	+	0.702906i	$0.751885\pi$
$488$	0	0
$489$	−1.02519	−0.0463608
$490$	0	0
$491$	39.9590	1.80333	0.901663	−	0.432440i	$-0.142347\pi$
$491$	39.9590	1.80333	0.901663	+	0.432440i	$0.142347\pi$
$492$	0	0
$493$	−9.34081	−0.420689
$494$	0	0
$495$	−5.86520	−0.263621
$496$	0	0
$497$	−2.44079	−0.109484
$498$	0	0
$499$	−16.0951	−0.720515	−0.360258	−	0.932853i	$-0.617311\pi$
$499$	−16.0951	−0.720515	−0.360258	+	0.932853i	$0.617311\pi$
$500$	0	0
$501$	1.30048	0.0581010
$502$	0	0
$503$	−5.97717	−0.266509	−0.133254	−	0.991082i	$-0.542543\pi$
$503$	−5.97717	−0.266509	−0.133254	+	0.991082i	$0.542543\pi$
$504$	0	0
$505$	−11.4587	−0.509907
$506$	0	0
$507$	−2.94447	−0.130769
$508$	0	0
$509$	12.5726	0.557269	0.278634	−	0.960397i	$-0.410118\pi$
$509$	12.5726	0.557269	0.278634	+	0.960397i	$0.410118\pi$
$510$	0	0
$511$	−26.2025	−1.15913
$512$	0	0
$513$	2.69220	0.118864
$514$	0	0
$515$	−0.759032	−0.0334469
$516$	0	0
$517$	14.3028	0.629036
$518$	0	0
$519$	−1.00610	−0.0441629
$520$	0	0
$521$	31.6233	1.38544	0.692722	−	0.721205i	$-0.256411\pi$
$521$	31.6233	1.38544	0.692722	+	0.721205i	$0.256411\pi$
$522$	0	0
$523$	−21.5405	−0.941900	−0.470950	−	0.882160i	$-0.656089\pi$
$523$	−21.5405	−0.941900	−0.470950	+	0.882160i	$0.656089\pi$
$524$	0	0
$525$	−1.08579	−0.0473876
$526$	0	0
$527$	8.56505	0.373099
$528$	0	0
$529$	54.8049	2.38282
$530$	0	0
$531$	−26.3824	−1.14490
$532$	0	0
$533$	0	0
$534$	0	0
$535$	19.6632	0.850115
$536$	0	0
$537$	−0.117475	−0.00506944
$538$	0	0
$539$	4.84463	0.208673
$540$	0	0
$541$	31.4927	1.35397	0.676987	−	0.735995i	$-0.263285\pi$
$541$	31.4927	1.35397	0.676987	+	0.735995i	$0.263285\pi$
$542$	0	0
$543$	−0.371784	−0.0159548
$544$	0	0
$545$	−4.50509	−0.192977
$546$	0	0
$547$	17.6861	0.756201	0.378100	−	0.925765i	$-0.376577\pi$
$547$	17.6861	0.756201	0.378100	+	0.925765i	$0.376577\pi$
$548$	0	0
$549$	−37.7774	−1.61230
$550$	0	0
$551$	8.55248	0.364348
$552$	0	0
$553$	2.88022	0.122479
$554$	0	0
$555$	0.221560	0.00940472
$556$	0	0
$557$	19.3296	0.819021	0.409511	−	0.912305i	$-0.365699\pi$
$557$	19.3296	0.819021	0.409511	+	0.912305i	$0.365699\pi$
$558$	0	0
$559$	−9.74235	−0.412057
$560$	0	0
$561$	0.822430	0.0347230
$562$	0	0
$563$	−35.7299	−1.50583	−0.752917	−	0.658115i	$-0.771354\pi$
$563$	−35.7299	−1.50583	−0.752917	+	0.658115i	$0.771354\pi$
$564$	0	0
$565$	−8.95293	−0.376653
$566$	0	0
$567$	−28.0494	−1.17797
$568$	0	0
$569$	4.65187	0.195016	0.0975082	−	0.995235i	$-0.468913\pi$
$569$	4.65187	0.195016	0.0975082	+	0.995235i	$0.468913\pi$
$570$	0	0
$571$	20.6791	0.865393	0.432697	−	0.901540i	$-0.357562\pi$
$571$	20.6791	0.865393	0.432697	+	0.901540i	$0.357562\pi$
$572$	0	0
$573$	−0.695813	−0.0290680
$574$	0	0
$575$	32.0238	1.33548
$576$	0	0
$577$	24.7808	1.03164	0.515820	−	0.856697i	$-0.327487\pi$
$577$	24.7808	1.03164	0.515820	+	0.856697i	$0.327487\pi$
$578$	0	0
$579$	−1.26484	−0.0525649
$580$	0	0
$581$	31.7641	1.31780
$582$	0	0
$583$	15.6701	0.648989
$584$	0	0
$585$	23.2079	0.959528
$586$	0	0
$587$	13.4027	0.553187	0.276594	−	0.960987i	$-0.410794\pi$
$587$	13.4027	0.553187	0.276594	+	0.960987i	$0.410794\pi$
$588$	0	0
$589$	−7.84219	−0.323132
$590$	0	0
$591$	1.48053	0.0609010
$592$	0	0
$593$	1.83911	0.0755233	0.0377616	−	0.999287i	$-0.487977\pi$
$593$	1.83911	0.0755233	0.0377616	+	0.999287i	$0.487977\pi$
$594$	0	0
$595$	18.9955	0.778740
$596$	0	0
$597$	−2.65285	−0.108574
$598$	0	0
$599$	−35.9457	−1.46870	−0.734352	−	0.678769i	$-0.762514\pi$
$599$	−35.9457	−1.46870	−0.734352	+	0.678769i	$0.762514\pi$
$600$	0	0
$601$	29.8858	1.21907	0.609533	−	0.792760i	$-0.291357\pi$
$601$	29.8858	1.21907	0.609533	+	0.792760i	$0.291357\pi$
$602$	0	0
$603$	28.9396	1.17851
$604$	0	0
$605$	−9.58671	−0.389755
$606$	0	0
$607$	4.25015	0.172508	0.0862541	−	0.996273i	$-0.472510\pi$
$607$	4.25015	0.172508	0.0862541	+	0.996273i	$0.472510\pi$
$608$	0	0
$609$	0.541264	0.0219331
$610$	0	0
$611$	−56.5944	−2.28956
$612$	0	0
$613$	−9.35184	−0.377717	−0.188859	−	0.982004i	$-0.560479\pi$
$613$	−9.35184	−0.377717	−0.188859	+	0.982004i	$0.560479\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.6019	0.507332	0.253666	−	0.967292i	$-0.418364\pi$
$617$	12.6019	0.507332	0.253666	+	0.967292i	$0.418364\pi$
$618$	0	0
$619$	−2.54018	−0.102098	−0.0510491	−	0.998696i	$-0.516257\pi$
$619$	−2.54018	−0.102098	−0.0510491	+	0.998696i	$0.516257\pi$
$620$	0	0
$621$	−5.02519	−0.201654
$622$	0	0
$623$	−20.2549	−0.811497
$624$	0	0
$625$	6.33331	0.253332
$626$	0	0
$627$	−0.753020	−0.0300727
$628$	0	0
$629$	10.2757	0.409720
$630$	0	0
$631$	48.0057	1.91108	0.955538	−	0.294867i	$-0.0952752\pi$
$631$	48.0057	1.91108	0.955538	+	0.294867i	$0.0952752\pi$
$632$	0	0
$633$	1.72580	0.0685945
$634$	0	0
$635$	17.4607	0.692908
$636$	0	0
$637$	−19.1696	−0.759528
$638$	0	0
$639$	−2.32123	−0.0918262
$640$	0	0
$641$	−1.02148	−0.0403461	−0.0201730	−	0.999797i	$-0.506422\pi$
$641$	−1.02148	−0.0403461	−0.0201730	+	0.999797i	$0.506422\pi$
$642$	0	0
$643$	8.40606	0.331503	0.165751	−	0.986168i	$-0.446995\pi$
$643$	8.40606	0.331503	0.165751	+	0.986168i	$0.446995\pi$
$644$	0	0
$645$	−0.163511	−0.00643823
$646$	0	0
$647$	5.86778	0.230686	0.115343	−	0.993326i	$-0.463203\pi$
$647$	5.86778	0.230686	0.115343	+	0.993326i	$0.463203\pi$
$648$	0	0
$649$	14.7808	0.580198
$650$	0	0
$651$	−0.496312	−0.0194520
$652$	0	0
$653$	−37.5749	−1.47042	−0.735209	−	0.677841i	$-0.762916\pi$
$653$	−37.5749	−1.47042	−0.735209	+	0.677841i	$0.762916\pi$
$654$	0	0
$655$	4.90355	0.191598
$656$	0	0
$657$	−24.9190	−0.972183
$658$	0	0
$659$	16.8360	0.655839	0.327919	−	0.944706i	$-0.393653\pi$
$659$	16.8360	0.655839	0.327919	+	0.944706i	$0.393653\pi$
$660$	0	0
$661$	4.84983	0.188637	0.0943183	−	0.995542i	$-0.469933\pi$
$661$	4.84983	0.188637	0.0943183	+	0.995542i	$0.469933\pi$
$662$	0	0
$663$	−3.25426	−0.126385
$664$	0	0
$665$	−17.3924	−0.674447
$666$	0	0
$667$	−15.9638	−0.618122
$668$	0	0
$669$	0.623651	0.0241117
$670$	0	0
$671$	21.1649	0.817062
$672$	0	0
$673$	−6.24065	−0.240559	−0.120280	−	0.992740i	$-0.538379\pi$
$673$	−6.24065	−0.240559	−0.120280	+	0.992740i	$0.538379\pi$
$674$	0	0
$675$	−2.06832	−0.0796098
$676$	0	0
$677$	−29.3566	−1.12827	−0.564134	−	0.825683i	$-0.690790\pi$
$677$	−29.3566	−1.12827	−0.564134	+	0.825683i	$0.690790\pi$
$678$	0	0
$679$	−26.1741	−1.00447
$680$	0	0
$681$	2.23360	0.0855918
$682$	0	0
$683$	25.6861	0.982849	0.491425	−	0.870920i	$-0.336476\pi$
$683$	25.6861	0.982849	0.491425	+	0.870920i	$0.336476\pi$
$684$	0	0
$685$	−4.79783	−0.183316
$686$	0	0
$687$	1.15542	0.0440820
$688$	0	0
$689$	−62.0046	−2.36219
$690$	0	0
$691$	−8.77128	−0.333675	−0.166838	−	0.985984i	$-0.553356\pi$
$691$	−8.77128	−0.333675	−0.166838	+	0.985984i	$0.553356\pi$
$692$	0	0
$693$	15.7626	0.598772
$694$	0	0
$695$	−14.0218	−0.531877
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−2.07706	−0.0785616
$700$	0	0
$701$	−15.1075	−0.570602	−0.285301	−	0.958438i	$-0.592094\pi$
$701$	−15.1075	−0.570602	−0.285301	+	0.958438i	$0.592094\pi$
$702$	0	0
$703$	−9.40850	−0.354848
$704$	0	0
$705$	−0.949851	−0.0357735
$706$	0	0
$707$	30.7951	1.15817
$708$	0	0
$709$	12.0132	0.451165	0.225583	−	0.974224i	$-0.427571\pi$
$709$	12.0132	0.451165	0.225583	+	0.974224i	$0.427571\pi$
$710$	0	0
$711$	2.73913	0.102725
$712$	0	0
$713$	14.6380	0.548198
$714$	0	0
$715$	−13.0023	−0.486259
$716$	0	0
$717$	2.05372	0.0766977
$718$	0	0
$719$	2.27778	0.0849468	0.0424734	−	0.999098i	$-0.486476\pi$
$719$	2.27778	0.0849468	0.0424734	+	0.999098i	$0.486476\pi$
$720$	0	0
$721$	2.03988	0.0759692
$722$	0	0
$723$	−1.20109	−0.0446689
$724$	0	0
$725$	−6.57056	−0.244025
$726$	0	0
$727$	−7.71952	−0.286301	−0.143151	−	0.989701i	$-0.545723\pi$
$727$	−7.71952	−0.286301	−0.143151	+	0.989701i	$0.545723\pi$
$728$	0	0
$729$	−26.5129	−0.981960
$730$	0	0
$731$	−7.58345	−0.280484
$732$	0	0
$733$	33.2753	1.22905	0.614526	−	0.788896i	$-0.289347\pi$
$733$	33.2753	1.22905	0.614526	+	0.788896i	$0.289347\pi$
$734$	0	0
$735$	−0.321733	−0.0118673
$736$	0	0
$737$	−16.2135	−0.597233
$738$	0	0
$739$	48.0057	1.76592	0.882959	−	0.469450i	$-0.155548\pi$
$739$	48.0057	1.76592	0.882959	+	0.469450i	$0.155548\pi$
$740$	0	0
$741$	2.97961	0.109459
$742$	0	0
$743$	43.1345	1.58245	0.791226	−	0.611524i	$-0.209443\pi$
$743$	43.1345	1.58245	0.791226	+	0.611524i	$0.209443\pi$
$744$	0	0
$745$	2.11793	0.0775948
$746$	0	0
$747$	30.2082	1.10526
$748$	0	0
$749$	−52.8445	−1.93090
$750$	0	0
$751$	6.46883	0.236051	0.118025	−	0.993011i	$-0.462344\pi$
$751$	6.46883	0.236051	0.118025	+	0.993011i	$0.462344\pi$
$752$	0	0
$753$	−1.97512	−0.0719775
$754$	0	0
$755$	24.9920	0.909550
$756$	0	0
$757$	39.5531	1.43758	0.718790	−	0.695227i	$-0.244696\pi$
$757$	39.5531	1.43758	0.718790	+	0.695227i	$0.244696\pi$
$758$	0	0
$759$	1.40557	0.0510189
$760$	0	0
$761$	28.5130	1.03360	0.516799	−	0.856107i	$-0.327124\pi$
$761$	28.5130	1.03360	0.516799	+	0.856107i	$0.327124\pi$
$762$	0	0
$763$	12.1073	0.438315
$764$	0	0
$765$	18.0650	0.653142
$766$	0	0
$767$	−58.4859	−2.11180
$768$	0	0
$769$	49.7184	1.79289	0.896445	−	0.443154i	$-0.146141\pi$
$769$	49.7184	1.79289	0.896445	+	0.443154i	$0.146141\pi$
$770$	0	0
$771$	1.40936	0.0507569
$772$	0	0
$773$	24.2539	0.872351	0.436175	−	0.899862i	$-0.356333\pi$
$773$	24.2539	0.872351	0.436175	+	0.899862i	$0.356333\pi$
$774$	0	0
$775$	6.02488	0.216420
$776$	0	0
$777$	−0.595439	−0.0213613
$778$	0	0
$779$	0	0
$780$	0	0
$781$	1.30048	0.0465347
$782$	0	0
$783$	1.03106	0.0368470
$784$	0	0
$785$	−15.3428	−0.547608
$786$	0	0
$787$	0.907346	0.0323434	0.0161717	−	0.999869i	$-0.494852\pi$
$787$	0.907346	0.0323434	0.0161717	+	0.999869i	$0.494852\pi$
$788$	0	0
$789$	1.75139	0.0623512
$790$	0	0
$791$	24.0608	0.855504
$792$	0	0
$793$	−83.7470	−2.97394
$794$	0	0
$795$	−1.04065	−0.0369082
$796$	0	0
$797$	−32.4934	−1.15098	−0.575488	−	0.817810i	$-0.695188\pi$
$797$	−32.4934	−1.15098	−0.575488	+	0.817810i	$0.695188\pi$
$798$	0	0
$799$	−44.0531	−1.55849
$800$	0	0
$801$	−19.2628	−0.680616
$802$	0	0
$803$	13.9610	0.492672
$804$	0	0
$805$	32.4641	1.14421
$806$	0	0
$807$	2.46213	0.0866711
$808$	0	0
$809$	2.07804	0.0730602	0.0365301	−	0.999333i	$-0.488370\pi$
$809$	2.07804	0.0730602	0.0365301	+	0.999333i	$0.488370\pi$
$810$	0	0
$811$	−13.7518	−0.482893	−0.241446	−	0.970414i	$-0.577622\pi$
$811$	−13.7518	−0.482893	−0.241446	+	0.970414i	$0.577622\pi$
$812$	0	0
$813$	1.66431	0.0583697
$814$	0	0
$815$	12.6162	0.441927
$816$	0	0
$817$	6.94344	0.242920
$818$	0	0
$819$	−62.3707	−2.17941
$820$	0	0
$821$	6.76979	0.236267	0.118134	−	0.992998i	$-0.462309\pi$
$821$	6.76979	0.236267	0.118134	+	0.992998i	$0.462309\pi$
$822$	0	0
$823$	−22.3575	−0.779335	−0.389667	−	0.920956i	$-0.627410\pi$
$823$	−22.3575	−0.779335	−0.389667	+	0.920956i	$0.627410\pi$
$824$	0	0
$825$	0.578518	0.0201414
$826$	0	0
$827$	−26.1160	−0.908143	−0.454072	−	0.890965i	$-0.650029\pi$
$827$	−26.1160	−0.908143	−0.454072	+	0.890965i	$0.650029\pi$
$828$	0	0
$829$	−19.7540	−0.686085	−0.343043	−	0.939320i	$-0.611458\pi$
$829$	−19.7540	−0.686085	−0.343043	+	0.939320i	$0.611458\pi$
$830$	0	0
$831$	−0.834144	−0.0289361
$832$	0	0
$833$	−14.9216	−0.517004
$834$	0	0
$835$	−16.0039	−0.553839
$836$	0	0
$837$	−0.945427	−0.0326787
$838$	0	0
$839$	−20.8188	−0.718745	−0.359373	−	0.933194i	$-0.617009\pi$
$839$	−20.8188	−0.718745	−0.359373	+	0.933194i	$0.617009\pi$
$840$	0	0
$841$	−25.7246	−0.887054
$842$	0	0
$843$	1.54256	0.0531287
$844$	0	0
$845$	36.2353	1.24653
$846$	0	0
$847$	25.7641	0.885265
$848$	0	0
$849$	1.13941	0.0391044
$850$	0	0
$851$	17.5617	0.602006
$852$	0	0
$853$	48.6666	1.66631	0.833157	−	0.553037i	$-0.186531\pi$
$853$	48.6666	1.66631	0.833157	+	0.553037i	$0.186531\pi$
$854$	0	0
$855$	−16.5404	−0.565670
$856$	0	0
$857$	−12.8515	−0.439001	−0.219500	−	0.975612i	$-0.570443\pi$
$857$	−12.8515	−0.439001	−0.219500	+	0.975612i	$0.570443\pi$
$858$	0	0
$859$	−1.60904	−0.0548998	−0.0274499	−	0.999623i	$-0.508739\pi$
$859$	−1.60904	−0.0548998	−0.0274499	+	0.999623i	$0.508739\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46.0494	−1.56754	−0.783770	−	0.621052i	$-0.786706\pi$
$863$	−46.0494	−1.56754	−0.783770	+	0.621052i	$0.786706\pi$
$864$	0	0
$865$	12.3813	0.420976
$866$	0	0
$867$	−0.916518	−0.0311266
$868$	0	0
$869$	−1.53461	−0.0520581
$870$	0	0
$871$	64.1549	2.17381
$872$	0	0
$873$	−24.8920	−0.842465
$874$	0	0
$875$	31.7641	1.07382
$876$	0	0
$877$	3.63392	0.122709	0.0613543	−	0.998116i	$-0.480458\pi$
$877$	3.63392	0.122709	0.0613543	+	0.998116i	$0.480458\pi$
$878$	0	0
$879$	−1.06995	−0.0360887
$880$	0	0
$881$	28.2426	0.951519	0.475759	−	0.879575i	$-0.342173\pi$
$881$	28.2426	0.951519	0.475759	+	0.879575i	$0.342173\pi$
$882$	0	0
$883$	4.23962	0.142674	0.0713372	−	0.997452i	$-0.477273\pi$
$883$	4.23962	0.142674	0.0713372	+	0.997452i	$0.477273\pi$
$884$	0	0
$885$	−0.981598	−0.0329961
$886$	0	0
$887$	46.4003	1.55797	0.778984	−	0.627043i	$-0.215735\pi$
$887$	46.4003	1.55797	0.778984	+	0.627043i	$0.215735\pi$
$888$	0	0
$889$	−46.9254	−1.57383
$890$	0	0
$891$	14.9450	0.500677
$892$	0	0
$893$	40.3352	1.34976
$894$	0	0
$895$	1.44568	0.0483237
$896$	0	0
$897$	−5.56166	−0.185698
$898$	0	0
$899$	−3.00339	−0.100169
$900$	0	0
$901$	−48.2644	−1.60792
$902$	0	0
$903$	0.439432	0.0146234
$904$	0	0
$905$	4.57526	0.152087
$906$	0	0
$907$	−27.0358	−0.897708	−0.448854	−	0.893605i	$-0.648168\pi$
$907$	−27.0358	−0.897708	−0.448854	+	0.893605i	$0.648168\pi$
$908$	0	0
$909$	29.2866	0.971376
$910$	0	0
$911$	−42.3500	−1.40312	−0.701559	−	0.712612i	$-0.747512\pi$
$911$	−42.3500	−1.40312	−0.701559	+	0.712612i	$0.747512\pi$
$912$	0	0
$913$	−16.9243	−0.560111
$914$	0	0
$915$	−1.40557	−0.0464666
$916$	0	0
$917$	−13.1782	−0.435183
$918$	0	0
$919$	−8.17400	−0.269635	−0.134818	−	0.990870i	$-0.543045\pi$
$919$	−8.17400	−0.269635	−0.134818	+	0.990870i	$0.543045\pi$
$920$	0	0
$921$	−0.818991	−0.0269867
$922$	0	0
$923$	−5.14582	−0.169377
$924$	0	0
$925$	7.22821	0.237662
$926$	0	0
$927$	1.93996	0.0637166
$928$	0	0
$929$	−42.2596	−1.38649	−0.693247	−	0.720700i	$-0.743820\pi$
$929$	−42.2596	−1.38649	−0.693247	+	0.720700i	$0.743820\pi$
$930$	0	0
$931$	13.6623	0.447764
$932$	0	0
$933$	3.01994	0.0988684
$934$	0	0
$935$	−10.1210	−0.330992
$936$	0	0
$937$	−1.16600	−0.0380917	−0.0190458	−	0.999819i	$-0.506063\pi$
$937$	−1.16600	−0.0380917	−0.0190458	+	0.999819i	$0.506063\pi$
$938$	0	0
$939$	0.210896	0.00688234
$940$	0	0
$941$	4.93864	0.160995	0.0804975	−	0.996755i	$-0.474349\pi$
$941$	4.93864	0.160995	0.0804975	+	0.996755i	$0.474349\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−2.09676	−0.0682077
$946$	0	0
$947$	25.3210	0.822822	0.411411	−	0.911450i	$-0.365036\pi$
$947$	25.3210	0.822822	0.411411	+	0.911450i	$0.365036\pi$
$948$	0	0
$949$	−55.2419	−1.79323
$950$	0	0
$951$	−1.74922	−0.0567224
$952$	0	0
$953$	−47.6143	−1.54238	−0.771189	−	0.636606i	$-0.780338\pi$
$953$	−47.6143	−1.54238	−0.771189	+	0.636606i	$0.780338\pi$
$954$	0	0
$955$	8.56283	0.277087
$956$	0	0
$957$	−0.288391	−0.00932235
$958$	0	0
$959$	12.8941	0.416371
$960$	0	0
$961$	−28.2460	−0.911163
$962$	0	0
$963$	−50.2559	−1.61947
$964$	0	0
$965$	15.5654	0.501067
$966$	0	0
$967$	25.9763	0.835342	0.417671	−	0.908598i	$-0.362847\pi$
$967$	25.9763	0.835342	0.417671	+	0.908598i	$0.362847\pi$
$968$	0	0
$969$	2.31933	0.0745076
$970$	0	0
$971$	55.8687	1.79291	0.896457	−	0.443132i	$-0.146133\pi$
$971$	55.8687	1.79291	0.896457	+	0.443132i	$0.146133\pi$
$972$	0	0
$973$	37.6833	1.20807
$974$	0	0
$975$	−2.28913	−0.0733107
$976$	0	0
$977$	42.5241	1.36047	0.680233	−	0.732996i	$-0.261879\pi$
$977$	42.5241	1.36047	0.680233	+	0.732996i	$0.261879\pi$
$978$	0	0
$979$	10.7920	0.344915
$980$	0	0
$981$	11.5143	0.367622
$982$	0	0
$983$	50.8977	1.62338	0.811692	−	0.584086i	$-0.198547\pi$
$983$	50.8977	1.62338	0.811692	+	0.584086i	$0.198547\pi$
$984$	0	0
$985$	−18.2198	−0.580530
$986$	0	0
$987$	2.55271	0.0812535
$988$	0	0
$989$	−12.9604	−0.412118
$990$	0	0
$991$	27.7147	0.880387	0.440194	−	0.897903i	$-0.354910\pi$
$991$	27.7147	0.880387	0.440194	+	0.897903i	$0.354910\pi$
$992$	0	0
$993$	0.961973	0.0305273
$994$	0	0
$995$	32.6466	1.03497
$996$	0	0
$997$	22.3480	0.707768	0.353884	−	0.935289i	$-0.384861\pi$
$997$	22.3480	0.707768	0.353884	+	0.935289i	$0.384861\pi$
$998$	0	0
$999$	−1.13426	−0.0358863

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6724.2.a.c.1.3		4
41.40	even	2		164.2.a.a.1.2	✓	4
123.122	odd	2		1476.2.a.g.1.3		4
164.163	odd	2		656.2.a.i.1.3		4
205.122	odd	4		4100.2.d.c.1149.4		8
205.163	odd	4		4100.2.d.c.1149.5		8
205.204	even	2		4100.2.a.c.1.3		4
287.286	odd	2		8036.2.a.i.1.3		4
328.163	odd	2		2624.2.a.y.1.2		4
328.245	even	2		2624.2.a.v.1.3		4
492.491	even	2		5904.2.a.bp.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.a.a.1.2	✓	4	41.40	even	2
656.2.a.i.1.3		4	164.163	odd	2
1476.2.a.g.1.3		4	123.122	odd	2
2624.2.a.v.1.3		4	328.245	even	2
2624.2.a.y.1.2		4	328.163	odd	2
4100.2.a.c.1.3		4	205.204	even	2
4100.2.d.c.1149.4		8	205.122	odd	4
4100.2.d.c.1149.5		8	205.163	odd	4
5904.2.a.bp.1.3		4	492.491	even	2
6724.2.a.c.1.3		4	1.1	even	1	trivial
8036.2.a.i.1.3		4	287.286	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:	\( N \)	\(=\)	\( 6724 = 2^{2} \cdot 41^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6724.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0950939 q^{3} +1.17025 q^{5} -3.14501 q^{7} -2.99096 q^{9} +O(q^{10})\)	\(q-0.0950939 q^{3} +1.17025 q^{5} -3.14501 q^{7} -2.99096 q^{9} +1.67570 q^{11} -6.63052 q^{13} -0.111283 q^{15} -5.16120 q^{17} +4.72562 q^{19} +0.299072 q^{21} -8.82071 q^{23} -3.63052 q^{25} +0.569704 q^{27} +1.80981 q^{29} -1.65951 q^{31} -0.159349 q^{33} -3.68044 q^{35} -1.99096 q^{37} +0.630522 q^{39} +1.46932 q^{43} -3.50016 q^{45} +8.53543 q^{47} +2.89112 q^{49} +0.490799 q^{51} +9.35139 q^{53} +1.96098 q^{55} -0.449377 q^{57} +8.82071 q^{59} +12.6305 q^{61} +9.40660 q^{63} -7.75935 q^{65} -9.67570 q^{67} +0.838796 q^{69} +0.776081 q^{71} +8.33145 q^{73} +0.345241 q^{75} -5.27009 q^{77} -0.915804 q^{79} +8.91870 q^{81} -10.0998 q^{83} -6.03988 q^{85} -0.172102 q^{87} +6.44033 q^{89} +20.8531 q^{91} +0.157809 q^{93} +5.53014 q^{95} +8.32241 q^{97} -5.01193 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 4 q^{5} + 12 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 4 * q^5 + 12 * q^9	\( 4 q - 2 q^{3} + 4 q^{5} + 12 q^{9} - 4 q^{11} + 10 q^{15} + 4 q^{17} - 6 q^{19} - 12 q^{23} + 12 q^{25} + 10 q^{27} + 4 q^{29} - 8 q^{31} - 20 q^{33} + 26 q^{35} + 16 q^{37} - 24 q^{39} + 4 q^{43} + 4 q^{45} + 6 q^{47} + 16 q^{49} - 4 q^{51} + 16 q^{53} + 2 q^{55} + 4 q^{57} + 12 q^{59} + 24 q^{61} + 10 q^{63} - 4 q^{65} - 28 q^{67} + 28 q^{69} + 2 q^{71} + 8 q^{73} - 30 q^{75} + 8 q^{77} + 18 q^{79} + 28 q^{81} - 12 q^{83} - 32 q^{85} + 44 q^{87} - 4 q^{89} + 36 q^{91} + 28 q^{93} - 14 q^{95} - 16 q^{97} - 58 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 4 * q^5 + 12 * q^9 - 4 * q^11 + 10 * q^15 + 4 * q^17 - 6 * q^19 - 12 * q^23 + 12 * q^25 + 10 * q^27 + 4 * q^29 - 8 * q^31 - 20 * q^33 + 26 * q^35 + 16 * q^37 - 24 * q^39 + 4 * q^43 + 4 * q^45 + 6 * q^47 + 16 * q^49 - 4 * q^51 + 16 * q^53 + 2 * q^55 + 4 * q^57 + 12 * q^59 + 24 * q^61 + 10 * q^63 - 4 * q^65 - 28 * q^67 + 28 * q^69 + 2 * q^71 + 8 * q^73 - 30 * q^75 + 8 * q^77 + 18 * q^79 + 28 * q^81 - 12 * q^83 - 32 * q^85 + 44 * q^87 - 4 * q^89 + 36 * q^91 + 28 * q^93 - 14 * q^95 - 16 * q^97 - 58 * q^99

Introduction

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