LMFDB - Embedded newform 4640.2.a.o.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4640,2,Mod(1,4640)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4640.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4640, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4640 = 2^{5} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4640.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4,0,0,0,12,0,0,0,16,0,0,0,4,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$37.0505865379$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{6}, \sqrt{26})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 16x^{2} + 25 $ x^4 - 16*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.32476$ of defining polynomial
Character	$\chi$	$=$	4640.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{3} -1.00000 q^{5} +2.44949 q^{7} +3.00000 q^{9} +5.09902 q^{11} +4.00000 q^{13} -2.44949 q^{15} -5.24500 q^{17} +2.44949 q^{19} +6.00000 q^{21} +5.09902 q^{23} +1.00000 q^{25} +1.00000 q^{29} +7.34847 q^{31} +12.4900 q^{33} -2.44949 q^{35} -9.24500 q^{37} +9.79796 q^{39} -8.00000 q^{41} -7.74855 q^{43} -3.00000 q^{45} +12.6475 q^{47} -1.00000 q^{49} -12.8476 q^{51} +8.00000 q^{53} -5.09902 q^{55} +6.00000 q^{57} -2.24945 q^{59} -14.4900 q^{61} +7.34847 q^{63} -4.00000 q^{65} -0.200040 q^{67} +12.4900 q^{69} -2.64953 q^{71} -0.755002 q^{73} +2.44949 q^{75} +12.4900 q^{77} +14.8970 q^{79} -9.00000 q^{81} +17.5465 q^{83} +5.24500 q^{85} +2.44949 q^{87} -6.00000 q^{89} +9.79796 q^{91} +18.0000 q^{93} -2.44949 q^{95} +11.2450 q^{97} +15.2971 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 12 q^{9} + 16 q^{13} + 4 q^{17} + 24 q^{21} + 4 q^{25} + 4 q^{29} - 12 q^{37} - 32 q^{41} - 12 q^{45} - 4 q^{49} + 32 q^{53} + 24 q^{57} - 8 q^{61} - 16 q^{65} - 28 q^{73} - 36 q^{81} - 4 q^{85}+ \cdots + 20 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 + 12 * q^9 + 16 * q^13 + 4 * q^17 + 24 * q^21 + 4 * q^25 + 4 * q^29 - 12 * q^37 - 32 * q^41 - 12 * q^45 - 4 * q^49 + 32 * q^53 + 24 * q^57 - 8 * q^61 - 16 * q^65 - 28 * q^73 - 36 * q^81 - 4 * q^85 - 24 * q^89 + 72 * q^93 + 20 * q^97

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
$3$	2.44949	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$3$	2.44949	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.44949	0.925820	0.462910	−	0.886405i	$-0.346805\pi$
$7$	2.44949	0.925820	0.462910	+	0.886405i	$0.346805\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	5.09902	1.53741	0.768706	−	0.639602i	$-0.220901\pi$
$11$	5.09902	1.53741	0.768706	+	0.639602i	$0.220901\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−2.44949	−0.632456
$16$	0	0
$17$	−5.24500	−1.27210	−0.636049	−	0.771648i	$-0.719433\pi$
$17$	−5.24500	−1.27210	−0.636049	+	0.771648i	$0.719433\pi$
$18$	0	0
$19$	2.44949	0.561951	0.280976	−	0.959715i	$-0.409342\pi$
$19$	2.44949	0.561951	0.280976	+	0.959715i	$0.409342\pi$
$20$	0	0
$21$	6.00000	1.30931
$22$	0	0
$23$	5.09902	1.06322	0.531610	−	0.846990i	$-0.321587\pi$
$23$	5.09902	1.06322	0.531610	+	0.846990i	$0.321587\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	7.34847	1.31982	0.659912	−	0.751343i	$-0.270594\pi$
$31$	7.34847	1.31982	0.659912	+	0.751343i	$0.270594\pi$
$32$	0	0
$33$	12.4900	2.17423
$34$	0	0
$35$	−2.44949	−0.414039
$36$	0	0
$37$	−9.24500	−1.51987	−0.759934	−	0.650000i	$-0.774769\pi$
$37$	−9.24500	−1.51987	−0.759934	+	0.650000i	$0.774769\pi$
$38$	0	0
$39$	9.79796	1.56893
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	−7.74855	−1.18164	−0.590821	−	0.806802i	$-0.701196\pi$
$43$	−7.74855	−1.18164	−0.590821	+	0.806802i	$0.701196\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	12.6475	1.84483	0.922416	−	0.386198i	$-0.126212\pi$
$47$	12.6475	1.84483	0.922416	+	0.386198i	$0.126212\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−12.8476	−1.79902
$52$	0	0
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	0	0
$55$	−5.09902	−0.687552
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	−2.24945	−0.292853	−0.146427	−	0.989222i	$-0.546777\pi$
$59$	−2.24945	−0.292853	−0.146427	+	0.989222i	$0.546777\pi$
$60$	0	0
$61$	−14.4900	−1.85525	−0.927627	−	0.373508i	$-0.878155\pi$
$61$	−14.4900	−1.85525	−0.927627	+	0.373508i	$0.878155\pi$
$62$	0	0
$63$	7.34847	0.925820
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−0.200040	−0.0244388	−0.0122194	−	0.999925i	$-0.503890\pi$
$67$	−0.200040	−0.0244388	−0.0122194	+	0.999925i	$0.503890\pi$
$68$	0	0
$69$	12.4900	1.50362
$70$	0	0
$71$	−2.64953	−0.314441	−0.157221	−	0.987563i	$-0.550253\pi$
$71$	−2.64953	−0.314441	−0.157221	+	0.987563i	$0.550253\pi$
$72$	0	0
$73$	−0.755002	−0.0883663	−0.0441832	−	0.999023i	$-0.514069\pi$
$73$	−0.755002	−0.0883663	−0.0441832	+	0.999023i	$0.514069\pi$
$74$	0	0
$75$	2.44949	0.282843
$76$	0	0
$77$	12.4900	1.42337
$78$	0	0
$79$	14.8970	1.67604	0.838021	−	0.545639i	$-0.183713\pi$
$79$	14.8970	1.67604	0.838021	+	0.545639i	$0.183713\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	17.5465	1.92598	0.962990	−	0.269538i	$-0.0868710\pi$
$83$	17.5465	1.92598	0.962990	+	0.269538i	$0.0868710\pi$
$84$	0	0
$85$	5.24500	0.568900
$86$	0	0
$87$	2.44949	0.262613
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	9.79796	1.02711
$92$	0	0
$93$	18.0000	1.86651
$94$	0	0
$95$	−2.44949	−0.251312
$96$	0	0
$97$	11.2450	1.14176	0.570878	−	0.821035i	$-0.306603\pi$
$97$	11.2450	1.14176	0.570878	+	0.821035i	$0.306603\pi$
$98$	0	0
$99$	15.2971	1.53741
$100$	0	0
$101$	−12.4900	−1.24280	−0.621401	−	0.783493i	$-0.713436\pi$
$101$	−12.4900	−1.24280	−0.621401	+	0.783493i	$0.713436\pi$
$102$	0	0
$103$	−9.99800	−0.985132	−0.492566	−	0.870275i	$-0.663941\pi$
$103$	−9.99800	−0.985132	−0.492566	+	0.870275i	$0.663941\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	0	0
$107$	−7.34847	−0.710403	−0.355202	−	0.934790i	$-0.615588\pi$
$107$	−7.34847	−0.710403	−0.355202	+	0.934790i	$0.615588\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	−22.6455	−2.14942
$112$	0	0
$113$	−17.7350	−1.66837	−0.834184	−	0.551486i	$-0.814061\pi$
$113$	−17.7350	−1.66837	−0.834184	+	0.551486i	$0.814061\pi$
$114$	0	0
$115$	−5.09902	−0.475486
$116$	0	0
$117$	12.0000	1.10940
$118$	0	0
$119$	−12.8476	−1.17773
$120$	0	0
$121$	15.0000	1.36364
$122$	0	0
$123$	−19.5959	−1.76690
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.2474	−1.08679	−0.543393	−	0.839479i	$-0.682861\pi$
$127$	−12.2474	−1.08679	−0.543393	+	0.839479i	$0.682861\pi$
$128$	0	0
$129$	−18.9800	−1.67110
$130$	0	0
$131$	−2.84957	−0.248968	−0.124484	−	0.992222i	$-0.539728\pi$
$131$	−2.84957	−0.248968	−0.124484	+	0.992222i	$0.539728\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.24500	−0.448110	−0.224055	−	0.974576i	$-0.571930\pi$
$137$	−5.24500	−0.448110	−0.224055	+	0.974576i	$0.571930\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	30.9800	2.60899
$142$	0	0
$143$	20.3961	1.70561
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−2.44949	−0.202031
$148$	0	0
$149$	−16.4900	−1.35091	−0.675457	−	0.737400i	$-0.736053\pi$
$149$	−16.4900	−1.35091	−0.675457	+	0.737400i	$0.736053\pi$
$150$	0	0
$151$	22.6455	1.84287	0.921433	−	0.388536i	$-0.127019\pi$
$151$	22.6455	1.84287	0.921433	+	0.388536i	$0.127019\pi$
$152$	0	0
$153$	−15.7350	−1.27210
$154$	0	0
$155$	−7.34847	−0.590243
$156$	0	0
$157$	4.75500	0.379490	0.189745	−	0.981833i	$-0.439234\pi$
$157$	4.75500	0.379490	0.189745	+	0.981833i	$0.439234\pi$
$158$	0	0
$159$	19.5959	1.55406
$160$	0	0
$161$	12.4900	0.984350
$162$	0	0
$163$	7.74855	0.606913	0.303457	−	0.952845i	$-0.401859\pi$
$163$	7.74855	0.606913	0.303457	+	0.952845i	$0.401859\pi$
$164$	0	0
$165$	−12.4900	−0.972345
$166$	0	0
$167$	−4.69894	−0.363615	−0.181807	−	0.983334i	$-0.558195\pi$
$167$	−4.69894	−0.363615	−0.181807	+	0.983334i	$0.558195\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	7.34847	0.561951
$172$	0	0
$173$	14.4900	1.10165	0.550827	−	0.834619i	$-0.314312\pi$
$173$	14.4900	1.10165	0.550827	+	0.834619i	$0.314312\pi$
$174$	0	0
$175$	2.44949	0.185164
$176$	0	0
$177$	−5.51000	−0.414157
$178$	0	0
$179$	14.6969	1.09850	0.549250	−	0.835658i	$-0.314913\pi$
$179$	14.6969	1.09850	0.549250	+	0.835658i	$0.314913\pi$
$180$	0	0
$181$	10.4900	0.779715	0.389858	−	0.920875i	$-0.372524\pi$
$181$	10.4900	0.779715	0.389858	+	0.920875i	$0.372524\pi$
$182$	0	0
$183$	−35.4931	−2.62373
$184$	0	0
$185$	9.24500	0.679706
$186$	0	0
$187$	−26.7443	−1.95574
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−25.0950	−1.81581	−0.907906	−	0.419174i	$-0.862320\pi$
$191$	−25.0950	−1.81581	−0.907906	+	0.419174i	$0.862320\pi$
$192$	0	0
$193$	19.7350	1.42056	0.710278	−	0.703921i	$-0.248569\pi$
$193$	19.7350	1.42056	0.710278	+	0.703921i	$0.248569\pi$
$194$	0	0
$195$	−9.79796	−0.701646
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	−7.94859	−0.563461	−0.281730	−	0.959494i	$-0.590908\pi$
$199$	−7.94859	−0.563461	−0.281730	+	0.959494i	$0.590908\pi$
$200$	0	0
$201$	−0.489996	−0.0345617
$202$	0	0
$203$	2.44949	0.171920
$204$	0	0
$205$	8.00000	0.558744
$206$	0	0
$207$	15.2971	1.06322
$208$	0	0
$209$	12.4900	0.863951
$210$	0	0
$211$	25.0950	1.72761	0.863806	−	0.503824i	$-0.168074\pi$
$211$	25.0950	1.72761	0.863806	+	0.503824i	$0.168074\pi$
$212$	0	0
$213$	−6.49000	−0.444687
$214$	0	0
$215$	7.74855	0.528447
$216$	0	0
$217$	18.0000	1.22192
$218$	0	0
$219$	−1.84937	−0.124969
$220$	0	0
$221$	−20.9800	−1.41127
$222$	0	0
$223$	9.99800	0.669516	0.334758	−	0.942304i	$-0.391346\pi$
$223$	9.99800	0.669516	0.334758	+	0.942304i	$0.391346\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	−9.99800	−0.663591	−0.331795	−	0.943351i	$-0.607654\pi$
$227$	−9.99800	−0.663591	−0.331795	+	0.943351i	$0.607654\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	30.5941	2.01295
$232$	0	0
$233$	−20.4900	−1.34234	−0.671172	−	0.741302i	$-0.734209\pi$
$233$	−20.4900	−1.34234	−0.671172	+	0.741302i	$0.734209\pi$
$234$	0	0
$235$	−12.6475	−0.825034
$236$	0	0
$237$	36.4900	2.37028
$238$	0	0
$239$	2.24945	0.145505	0.0727524	−	0.997350i	$-0.476822\pi$
$239$	2.24945	0.145505	0.0727524	+	0.997350i	$0.476822\pi$
$240$	0	0
$241$	22.9800	1.48027	0.740136	−	0.672458i	$-0.234761\pi$
$241$	22.9800	1.48027	0.740136	+	0.672458i	$0.234761\pi$
$242$	0	0
$243$	−22.0454	−1.41421
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	9.79796	0.623429
$248$	0	0
$249$	42.9800	2.72375
$250$	0	0
$251$	−9.59792	−0.605815	−0.302908	−	0.953020i	$-0.597957\pi$
$251$	−9.59792	−0.605815	−0.302908	+	0.953020i	$0.597957\pi$
$252$	0	0
$253$	26.0000	1.63461
$254$	0	0
$255$	12.8476	0.804546
$256$	0	0
$257$	−10.4900	−0.654348	−0.327174	−	0.944964i	$-0.606096\pi$
$257$	−10.4900	−0.654348	−0.327174	+	0.944964i	$0.606096\pi$
$258$	0	0
$259$	−22.6455	−1.40712
$260$	0	0
$261$	3.00000	0.185695
$262$	0	0
$263$	−7.74855	−0.477796	−0.238898	−	0.971045i	$-0.576786\pi$
$263$	−7.74855	−0.477796	−0.238898	+	0.971045i	$0.576786\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	0	0
$267$	−14.6969	−0.899438
$268$	0	0
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−15.2971	−0.929230	−0.464615	−	0.885513i	$-0.653807\pi$
$271$	−15.2971	−0.929230	−0.464615	+	0.885513i	$0.653807\pi$
$272$	0	0
$273$	24.0000	1.45255
$274$	0	0
$275$	5.09902	0.307482
$276$	0	0
$277$	20.9800	1.26057	0.630283	−	0.776366i	$-0.282939\pi$
$277$	20.9800	1.26057	0.630283	+	0.776366i	$0.282939\pi$
$278$	0	0
$279$	22.0454	1.31982
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	25.0950	1.49174	0.745872	−	0.666089i	$-0.232033\pi$
$283$	25.0950	1.49174	0.745872	+	0.666089i	$0.232033\pi$
$284$	0	0
$285$	−6.00000	−0.355409
$286$	0	0
$287$	−19.5959	−1.15671
$288$	0	0
$289$	10.5100	0.618236
$290$	0	0
$291$	27.5445	1.61469
$292$	0	0
$293$	3.24500	0.189575	0.0947874	−	0.995498i	$-0.469783\pi$
$293$	3.24500	0.189575	0.0947874	+	0.995498i	$0.469783\pi$
$294$	0	0
$295$	2.24945	0.130968
$296$	0	0
$297$	0	0
$298$	0	0
$299$	20.3961	1.17954
$300$	0	0
$301$	−18.9800	−1.09399
$302$	0	0
$303$	−30.5941	−1.75759
$304$	0	0
$305$	14.4900	0.829695
$306$	0	0
$307$	−12.2474	−0.698999	−0.349499	−	0.936937i	$-0.613648\pi$
$307$	−12.2474	−0.698999	−0.349499	+	0.936937i	$0.613648\pi$
$308$	0	0
$309$	−24.4900	−1.39319
$310$	0	0
$311$	−2.44949	−0.138898	−0.0694489	−	0.997586i	$-0.522124\pi$
$311$	−2.44949	−0.138898	−0.0694489	+	0.997586i	$0.522124\pi$
$312$	0	0
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	0	0
$315$	−7.34847	−0.414039
$316$	0	0
$317$	14.7550	0.828723	0.414362	−	0.910112i	$-0.364005\pi$
$317$	14.7550	0.828723	0.414362	+	0.910112i	$0.364005\pi$
$318$	0	0
$319$	5.09902	0.285490
$320$	0	0
$321$	−18.0000	−1.00466
$322$	0	0
$323$	−12.8476	−0.714858
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	39.1918	2.16731
$328$	0	0
$329$	30.9800	1.70798
$330$	0	0
$331$	−6.94839	−0.381918	−0.190959	−	0.981598i	$-0.561160\pi$
$331$	−6.94839	−0.381918	−0.190959	+	0.981598i	$0.561160\pi$
$332$	0	0
$333$	−27.7350	−1.51987
$334$	0	0
$335$	0.200040	0.0109294
$336$	0	0
$337$	−25.7350	−1.40187	−0.700937	−	0.713223i	$-0.747235\pi$
$337$	−25.7350	−1.40187	−0.700937	+	0.713223i	$0.747235\pi$
$338$	0	0
$339$	−43.4417	−2.35943
$340$	0	0
$341$	37.4700	2.02911
$342$	0	0
$343$	−19.5959	−1.05808
$344$	0	0
$345$	−12.4900	−0.672439
$346$	0	0
$347$	−11.8474	−0.636000	−0.318000	−	0.948091i	$-0.603011\pi$
$347$	−11.8474	−0.636000	−0.318000	+	0.948091i	$0.603011\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.9800	1.75535	0.877674	−	0.479258i	$-0.159094\pi$
$353$	32.9800	1.75535	0.877674	+	0.479258i	$0.159094\pi$
$354$	0	0
$355$	2.64953	0.140622
$356$	0	0
$357$	−31.4700	−1.66557
$358$	0	0
$359$	−2.44949	−0.129279	−0.0646396	−	0.997909i	$-0.520590\pi$
$359$	−2.44949	−0.129279	−0.0646396	+	0.997909i	$0.520590\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	36.7423	1.92847
$364$	0	0
$365$	0.755002	0.0395186
$366$	0	0
$367$	8.14863	0.425355	0.212677	−	0.977122i	$-0.431782\pi$
$367$	8.14863	0.425355	0.212677	+	0.977122i	$0.431782\pi$
$368$	0	0
$369$	−24.0000	−1.24939
$370$	0	0
$371$	19.5959	1.01737
$372$	0	0
$373$	3.51000	0.181741	0.0908706	−	0.995863i	$-0.471035\pi$
$373$	3.51000	0.181741	0.0908706	+	0.995863i	$0.471035\pi$
$374$	0	0
$375$	−2.44949	−0.126491
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	−29.9940	−1.54069	−0.770344	−	0.637628i	$-0.779916\pi$
$379$	−29.9940	−1.54069	−0.770344	+	0.637628i	$0.779916\pi$
$380$	0	0
$381$	−30.0000	−1.53695
$382$	0	0
$383$	17.5465	0.896585	0.448292	−	0.893887i	$-0.352032\pi$
$383$	17.5465	0.896585	0.448292	+	0.893887i	$0.352032\pi$
$384$	0	0
$385$	−12.4900	−0.636549
$386$	0	0
$387$	−23.2456	−1.18164
$388$	0	0
$389$	20.0000	1.01404	0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000	1.01404	0.507020	+	0.861934i	$0.330747\pi$
$390$	0	0
$391$	−26.7443	−1.35252
$392$	0	0
$393$	−6.97999	−0.352094
$394$	0	0
$395$	−14.8970	−0.749548
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	0	0
$399$	14.6969	0.735767
$400$	0	0
$401$	−26.4900	−1.32285	−0.661424	−	0.750013i	$-0.730047\pi$
$401$	−26.4900	−1.32285	−0.661424	+	0.750013i	$0.730047\pi$
$402$	0	0
$403$	29.3939	1.46421
$404$	0	0
$405$	9.00000	0.447214
$406$	0	0
$407$	−47.1404	−2.33666
$408$	0	0
$409$	1.51000	0.0746649	0.0373324	−	0.999303i	$-0.488114\pi$
$409$	1.51000	0.0746649	0.0373324	+	0.999303i	$0.488114\pi$
$410$	0	0
$411$	−12.8476	−0.633724
$412$	0	0
$413$	−5.51000	−0.271130
$414$	0	0
$415$	−17.5465	−0.861324
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.64953	0.129438	0.0647190	−	0.997904i	$-0.479385\pi$
$419$	2.64953	0.129438	0.0647190	+	0.997904i	$0.479385\pi$
$420$	0	0
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	0	0
$423$	37.9426	1.84483
$424$	0	0
$425$	−5.24500	−0.254420
$426$	0	0
$427$	−35.4931	−1.71763
$428$	0	0
$429$	49.9600	2.41209
$430$	0	0
$431$	−14.2969	−0.688655	−0.344328	−	0.938850i	$-0.611893\pi$
$431$	−14.2969	−0.688655	−0.344328	+	0.938850i	$0.611893\pi$
$432$	0	0
$433$	−39.7350	−1.90954	−0.954771	−	0.297342i	$-0.903900\pi$
$433$	−39.7350	−1.90954	−0.954771	+	0.297342i	$0.903900\pi$
$434$	0	0
$435$	−2.44949	−0.117444
$436$	0	0
$437$	12.4900	0.597478
$438$	0	0
$439$	32.8436	1.56754	0.783769	−	0.621053i	$-0.213295\pi$
$439$	32.8436	1.56754	0.783769	+	0.621053i	$0.213295\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−7.74855	−0.368145	−0.184072	−	0.982913i	$-0.558928\pi$
$443$	−7.74855	−0.368145	−0.184072	+	0.982913i	$0.558928\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	−40.3921	−1.91048
$448$	0	0
$449$	−6.49000	−0.306282	−0.153141	−	0.988204i	$-0.548939\pi$
$449$	−6.49000	−0.306282	−0.153141	+	0.988204i	$0.548939\pi$
$450$	0	0
$451$	−40.7922	−1.92083
$452$	0	0
$453$	55.4700	2.60621
$454$	0	0
$455$	−9.79796	−0.459335
$456$	0	0
$457$	12.9800	0.607178	0.303589	−	0.952803i	$-0.401815\pi$
$457$	12.9800	0.607178	0.303589	+	0.952803i	$0.401815\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.51000	−0.163477	−0.0817386	−	0.996654i	$-0.526047\pi$
$461$	−3.51000	−0.163477	−0.0817386	+	0.996654i	$0.526047\pi$
$462$	0	0
$463$	−19.3959	−0.901403	−0.450701	−	0.892675i	$-0.648826\pi$
$463$	−19.3959	−0.901403	−0.450701	+	0.892675i	$0.648826\pi$
$464$	0	0
$465$	−18.0000	−0.834730
$466$	0	0
$467$	2.44949	0.113349	0.0566744	−	0.998393i	$-0.481950\pi$
$467$	2.44949	0.113349	0.0566744	+	0.998393i	$0.481950\pi$
$468$	0	0
$469$	−0.489996	−0.0226259
$470$	0	0
$471$	11.6473	0.536681
$472$	0	0
$473$	−39.5100	−1.81667
$474$	0	0
$475$	2.44949	0.112390
$476$	0	0
$477$	24.0000	1.09888
$478$	0	0
$479$	25.4951	1.16490	0.582450	−	0.812866i	$-0.302094\pi$
$479$	25.4951	1.16490	0.582450	+	0.812866i	$0.302094\pi$
$480$	0	0
$481$	−36.9800	−1.68614
$482$	0	0
$483$	30.5941	1.39208
$484$	0	0
$485$	−11.2450	−0.510609
$486$	0	0
$487$	−12.2474	−0.554985	−0.277492	−	0.960728i	$-0.589503\pi$
$487$	−12.2474	−0.554985	−0.277492	+	0.960728i	$0.589503\pi$
$488$	0	0
$489$	18.9800	0.858305
$490$	0	0
$491$	7.74855	0.349687	0.174844	−	0.984596i	$-0.444058\pi$
$491$	7.74855	0.349687	0.174844	+	0.984596i	$0.444058\pi$
$492$	0	0
$493$	−5.24500	−0.236223
$494$	0	0
$495$	−15.2971	−0.687552
$496$	0	0
$497$	−6.49000	−0.291116
$498$	0	0
$499$	29.3939	1.31585	0.657925	−	0.753083i	$-0.271434\pi$
$499$	29.3939	1.31585	0.657925	+	0.753083i	$0.271434\pi$
$500$	0	0
$501$	−11.5100	−0.514229
$502$	0	0
$503$	−37.1424	−1.65610	−0.828049	−	0.560655i	$-0.810549\pi$
$503$	−37.1424	−1.65610	−0.828049	+	0.560655i	$0.810549\pi$
$504$	0	0
$505$	12.4900	0.555798
$506$	0	0
$507$	7.34847	0.326357
$508$	0	0
$509$	−36.9800	−1.63911	−0.819555	−	0.573001i	$-0.805779\pi$
$509$	−36.9800	−1.63911	−0.819555	+	0.573001i	$0.805779\pi$
$510$	0	0
$511$	−1.84937	−0.0818113
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9.99800	0.440564
$516$	0	0
$517$	64.4900	2.83627
$518$	0	0
$519$	35.4931	1.55797
$520$	0	0
$521$	−24.9800	−1.09439	−0.547197	−	0.837004i	$-0.684305\pi$
$521$	−24.9800	−1.09439	−0.547197	+	0.837004i	$0.684305\pi$
$522$	0	0
$523$	−30.3941	−1.32904	−0.664520	−	0.747270i	$-0.731364\pi$
$523$	−30.3941	−1.32904	−0.664520	+	0.747270i	$0.731364\pi$
$524$	0	0
$525$	6.00000	0.261861
$526$	0	0
$527$	−38.5427	−1.67895
$528$	0	0
$529$	3.00000	0.130435
$530$	0	0
$531$	−6.74835	−0.292853
$532$	0	0
$533$	−32.0000	−1.38607
$534$	0	0
$535$	7.34847	0.317702
$536$	0	0
$537$	36.0000	1.55351
$538$	0	0
$539$	−5.09902	−0.219630
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	25.6951	1.10268
$544$	0	0
$545$	−16.0000	−0.685365
$546$	0	0
$547$	−6.94839	−0.297092	−0.148546	−	0.988906i	$-0.547459\pi$
$547$	−6.94839	−0.297092	−0.148546	+	0.988906i	$0.547459\pi$
$548$	0	0
$549$	−43.4700	−1.85525
$550$	0	0
$551$	2.44949	0.104352
$552$	0	0
$553$	36.4900	1.55171
$554$	0	0
$555$	22.6455	0.961249
$556$	0	0
$557$	7.51000	0.318209	0.159105	−	0.987262i	$-0.449139\pi$
$557$	7.51000	0.318209	0.159105	+	0.987262i	$0.449139\pi$
$558$	0	0
$559$	−30.9942	−1.31091
$560$	0	0
$561$	−65.5100	−2.76583
$562$	0	0
$563$	21.6453	0.912242	0.456121	−	0.889918i	$-0.349238\pi$
$563$	21.6453	0.912242	0.456121	+	0.889918i	$0.349238\pi$
$564$	0	0
$565$	17.7350	0.746117
$566$	0	0
$567$	−22.0454	−0.925820
$568$	0	0
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	25.6951	1.07531	0.537654	−	0.843166i	$-0.319311\pi$
$571$	25.6951	1.07531	0.537654	+	0.843166i	$0.319311\pi$
$572$	0	0
$573$	−61.4700	−2.56795
$574$	0	0
$575$	5.09902	0.212644
$576$	0	0
$577$	−25.2450	−1.05096	−0.525482	−	0.850805i	$-0.676115\pi$
$577$	−25.2450	−1.05096	−0.525482	+	0.850805i	$0.676115\pi$
$578$	0	0
$579$	48.3407	2.00897
$580$	0	0
$581$	42.9800	1.78311
$582$	0	0
$583$	40.7922	1.68944
$584$	0	0
$585$	−12.0000	−0.496139
$586$	0	0
$587$	−14.8970	−0.614864	−0.307432	−	0.951570i	$-0.599470\pi$
$587$	−14.8970	−0.614864	−0.307432	+	0.951570i	$0.599470\pi$
$588$	0	0
$589$	18.0000	0.741677
$590$	0	0
$591$	4.89898	0.201517
$592$	0	0
$593$	−18.4900	−0.759293	−0.379647	−	0.925132i	$-0.623954\pi$
$593$	−18.4900	−0.759293	−0.379647	+	0.925132i	$0.623954\pi$
$594$	0	0
$595$	12.8476	0.526699
$596$	0	0
$597$	−19.4700	−0.796854
$598$	0	0
$599$	−25.4951	−1.04170	−0.520851	−	0.853648i	$-0.674385\pi$
$599$	−25.4951	−1.04170	−0.520851	+	0.853648i	$0.674385\pi$
$600$	0	0
$601$	32.9800	1.34528	0.672641	−	0.739969i	$-0.265160\pi$
$601$	32.9800	1.34528	0.672641	+	0.739969i	$0.265160\pi$
$602$	0	0
$603$	−0.600120	−0.0244388
$604$	0	0
$605$	−15.0000	−0.609837
$606$	0	0
$607$	−37.1424	−1.50756	−0.753782	−	0.657124i	$-0.771773\pi$
$607$	−37.1424	−1.50756	−0.753782	+	0.657124i	$0.771773\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	50.5901	2.04666
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	19.5959	0.790184
$616$	0	0
$617$	28.2250	1.13630	0.568148	−	0.822927i	$-0.307660\pi$
$617$	28.2250	1.13630	0.568148	+	0.822927i	$0.307660\pi$
$618$	0	0
$619$	−15.2971	−0.614841	−0.307420	−	0.951574i	$-0.599466\pi$
$619$	−15.2971	−0.614841	−0.307420	+	0.951574i	$0.599466\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−14.6969	−0.588820
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	30.5941	1.22181
$628$	0	0
$629$	48.4900	1.93342
$630$	0	0
$631$	15.8972	0.632857	0.316428	−	0.948616i	$-0.397516\pi$
$631$	15.8972	0.632857	0.316428	+	0.948616i	$0.397516\pi$
$632$	0	0
$633$	61.4700	2.44321
$634$	0	0
$635$	12.2474	0.486025
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	−7.94859	−0.314441
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	24.2949	0.958096	0.479048	−	0.877789i	$-0.340982\pi$
$643$	24.2949	0.958096	0.479048	+	0.877789i	$0.340982\pi$
$644$	0	0
$645$	18.9800	0.747336
$646$	0	0
$647$	−6.94839	−0.273169	−0.136585	−	0.990628i	$-0.543613\pi$
$647$	−6.94839	−0.273169	−0.136585	+	0.990628i	$0.543613\pi$
$648$	0	0
$649$	−11.4700	−0.450236
$650$	0	0
$651$	44.0908	1.72806
$652$	0	0
$653$	−0.755002	−0.0295455	−0.0147728	−	0.999891i	$-0.504702\pi$
$653$	−0.755002	−0.0295455	−0.0147728	+	0.999891i	$0.504702\pi$
$654$	0	0
$655$	2.84957	0.111342
$656$	0	0
$657$	−2.26501	−0.0883663
$658$	0	0
$659$	−11.8474	−0.461508	−0.230754	−	0.973012i	$-0.574119\pi$
$659$	−11.8474	−0.461508	−0.230754	+	0.973012i	$0.574119\pi$
$660$	0	0
$661$	−48.9800	−1.90510	−0.952550	−	0.304381i	$-0.901550\pi$
$661$	−48.9800	−1.90510	−0.952550	+	0.304381i	$0.901550\pi$
$662$	0	0
$663$	−51.3903	−1.99583
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	5.09902	0.197435
$668$	0	0
$669$	24.4900	0.946838
$670$	0	0
$671$	−73.8848	−2.85229
$672$	0	0
$673$	1.02001	0.0393184	0.0196592	−	0.999807i	$-0.493742\pi$
$673$	1.02001	0.0393184	0.0196592	+	0.999807i	$0.493742\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.24500	0.278448	0.139224	−	0.990261i	$-0.455539\pi$
$677$	7.24500	0.278448	0.139224	+	0.990261i	$0.455539\pi$
$678$	0	0
$679$	27.5445	1.05706
$680$	0	0
$681$	−24.4900	−0.938459
$682$	0	0
$683$	19.3959	0.742163	0.371081	−	0.928600i	$-0.378987\pi$
$683$	19.3959	0.742163	0.371081	+	0.928600i	$0.378987\pi$
$684$	0	0
$685$	5.24500	0.200401
$686$	0	0
$687$	34.2929	1.30835
$688$	0	0
$689$	32.0000	1.21910
$690$	0	0
$691$	27.9446	1.06306	0.531531	−	0.847039i	$-0.321617\pi$
$691$	27.9446	1.06306	0.531531	+	0.847039i	$0.321617\pi$
$692$	0	0
$693$	37.4700	1.42337
$694$	0	0
$695$	0	0
$696$	0	0
$697$	41.9600	1.58935
$698$	0	0
$699$	−50.1900	−1.89836
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	−22.6455	−0.854092
$704$	0	0
$705$	−30.9800	−1.16677
$706$	0	0
$707$	−30.5941	−1.15061
$708$	0	0
$709$	24.0000	0.901339	0.450669	−	0.892691i	$-0.351185\pi$
$709$	24.0000	0.901339	0.450669	+	0.892691i	$0.351185\pi$
$710$	0	0
$711$	44.6909	1.67604
$712$	0	0
$713$	37.4700	1.40326
$714$	0	0
$715$	−20.3961	−0.762770
$716$	0	0
$717$	5.51000	0.205775
$718$	0	0
$719$	27.5445	1.02724	0.513618	−	0.858019i	$-0.328305\pi$
$719$	27.5445	1.02724	0.513618	+	0.858019i	$0.328305\pi$
$720$	0	0
$721$	−24.4900	−0.912055
$722$	0	0
$723$	56.2893	2.09342
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	37.1424	1.37754	0.688768	−	0.724982i	$-0.258152\pi$
$727$	37.1424	1.37754	0.688768	+	0.724982i	$0.258152\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	40.6411	1.50317
$732$	0	0
$733$	−37.2450	−1.37568	−0.687838	−	0.725865i	$-0.741440\pi$
$733$	−37.2450	−1.37568	−0.687838	+	0.725865i	$0.741440\pi$
$734$	0	0
$735$	2.44949	0.0903508
$736$	0	0
$737$	−1.02001	−0.0375725
$738$	0	0
$739$	−2.84957	−0.104823	−0.0524116	−	0.998626i	$-0.516691\pi$
$739$	−2.84957	−0.104823	−0.0524116	+	0.998626i	$0.516691\pi$
$740$	0	0
$741$	24.0000	0.881662
$742$	0	0
$743$	−2.04941	−0.0751855	−0.0375928	−	0.999293i	$-0.511969\pi$
$743$	−2.04941	−0.0751855	−0.0375928	+	0.999293i	$0.511969\pi$
$744$	0	0
$745$	16.4900	0.604147
$746$	0	0
$747$	52.6395	1.92598
$748$	0	0
$749$	−18.0000	−0.657706
$750$	0	0
$751$	−7.74855	−0.282749	−0.141374	−	0.989956i	$-0.545152\pi$
$751$	−7.74855	−0.282749	−0.141374	+	0.989956i	$0.545152\pi$
$752$	0	0
$753$	−23.5100	−0.856752
$754$	0	0
$755$	−22.6455	−0.824155
$756$	0	0
$757$	25.7350	0.935354	0.467677	−	0.883899i	$-0.345091\pi$
$757$	25.7350	0.935354	0.467677	+	0.883899i	$0.345091\pi$
$758$	0	0
$759$	63.6867	2.31168
$760$	0	0
$761$	−47.9600	−1.73855	−0.869274	−	0.494331i	$-0.835413\pi$
$761$	−47.9600	−1.73855	−0.869274	+	0.494331i	$0.835413\pi$
$762$	0	0
$763$	39.1918	1.41884
$764$	0	0
$765$	15.7350	0.568900
$766$	0	0
$767$	−8.99780	−0.324892
$768$	0	0
$769$	−4.49000	−0.161913	−0.0809567	−	0.996718i	$-0.525798\pi$
$769$	−4.49000	−0.161913	−0.0809567	+	0.996718i	$0.525798\pi$
$770$	0	0
$771$	−25.6951	−0.925388
$772$	0	0
$773$	−15.7350	−0.565948	−0.282974	−	0.959128i	$-0.591321\pi$
$773$	−15.7350	−0.565948	−0.282974	+	0.959128i	$0.591321\pi$
$774$	0	0
$775$	7.34847	0.263965
$776$	0	0
$777$	−55.4700	−1.98997
$778$	0	0
$779$	−19.5959	−0.702097
$780$	0	0
$781$	−13.5100	−0.483426
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.75500	−0.169713
$786$	0	0
$787$	47.3405	1.68751	0.843753	−	0.536732i	$-0.180341\pi$
$787$	47.3405	1.68751	0.843753	+	0.536732i	$0.180341\pi$
$788$	0	0
$789$	−18.9800	−0.675706
$790$	0	0
$791$	−43.4417	−1.54461
$792$	0	0
$793$	−57.9600	−2.05822
$794$	0	0
$795$	−19.5959	−0.694996
$796$	0	0
$797$	−35.2450	−1.24844	−0.624221	−	0.781248i	$-0.714583\pi$
$797$	−35.2450	−1.24844	−0.624221	+	0.781248i	$0.714583\pi$
$798$	0	0
$799$	−66.3363	−2.34681
$800$	0	0
$801$	−18.0000	−0.635999
$802$	0	0
$803$	−3.84977	−0.135855
$804$	0	0
$805$	−12.4900	−0.440215
$806$	0	0
$807$	−58.7878	−2.06943
$808$	0	0
$809$	−46.9800	−1.65173	−0.825864	−	0.563869i	$-0.809312\pi$
$809$	−46.9800	−1.65173	−0.825864	+	0.563869i	$0.809312\pi$
$810$	0	0
$811$	0.400080	0.0140487	0.00702436	−	0.999975i	$-0.497764\pi$
$811$	0.400080	0.0140487	0.00702436	+	0.999975i	$0.497764\pi$
$812$	0	0
$813$	−37.4700	−1.31413
$814$	0	0
$815$	−7.74855	−0.271420
$816$	0	0
$817$	−18.9800	−0.664026
$818$	0	0
$819$	29.3939	1.02711
$820$	0	0
$821$	−45.4700	−1.58691	−0.793457	−	0.608627i	$-0.791721\pi$
$821$	−45.4700	−1.58691	−0.793457	+	0.608627i	$0.791721\pi$
$822$	0	0
$823$	17.5465	0.611633	0.305816	−	0.952091i	$-0.401071\pi$
$823$	17.5465	0.611633	0.305816	+	0.952091i	$0.401071\pi$
$824$	0	0
$825$	12.4900	0.434846
$826$	0	0
$827$	22.8456	0.794418	0.397209	−	0.917728i	$-0.369979\pi$
$827$	22.8456	0.794418	0.397209	+	0.917728i	$0.369979\pi$
$828$	0	0
$829$	44.9800	1.56222	0.781110	−	0.624394i	$-0.214654\pi$
$829$	44.9800	1.56222	0.781110	+	0.624394i	$0.214654\pi$
$830$	0	0
$831$	51.3903	1.78271
$832$	0	0
$833$	5.24500	0.181728
$834$	0	0
$835$	4.69894	0.162614
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24.2949	0.838752	0.419376	−	0.907813i	$-0.362249\pi$
$839$	24.2949	0.838752	0.419376	+	0.907813i	$0.362249\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	29.3939	1.01238
$844$	0	0
$845$	−3.00000	−0.103203
$846$	0	0
$847$	36.7423	1.26248
$848$	0	0
$849$	61.4700	2.10964
$850$	0	0
$851$	−47.1404	−1.61595
$852$	0	0
$853$	3.73499	0.127884	0.0639419	−	0.997954i	$-0.479633\pi$
$853$	3.73499	0.127884	0.0639419	+	0.997954i	$0.479633\pi$
$854$	0	0
$855$	−7.34847	−0.251312
$856$	0	0
$857$	−43.4700	−1.48491	−0.742453	−	0.669898i	$-0.766338\pi$
$857$	−43.4700	−1.48491	−0.742453	+	0.669898i	$0.766338\pi$
$858$	0	0
$859$	−39.3919	−1.34403	−0.672017	−	0.740536i	$-0.734572\pi$
$859$	−39.3919	−1.34403	−0.672017	+	0.740536i	$0.734572\pi$
$860$	0	0
$861$	−48.0000	−1.63584
$862$	0	0
$863$	−2.84957	−0.0970005	−0.0485002	−	0.998823i	$-0.515444\pi$
$863$	−2.84957	−0.0970005	−0.0485002	+	0.998823i	$0.515444\pi$
$864$	0	0
$865$	−14.4900	−0.492675
$866$	0	0
$867$	25.7441	0.874317
$868$	0	0
$869$	75.9600	2.57677
$870$	0	0
$871$	−0.800160	−0.0271124
$872$	0	0
$873$	33.7350	1.14176
$874$	0	0
$875$	−2.44949	−0.0828079
$876$	0	0
$877$	−40.9800	−1.38380	−0.691898	−	0.721995i	$-0.743225\pi$
$877$	−40.9800	−1.38380	−0.691898	+	0.721995i	$0.743225\pi$
$878$	0	0
$879$	7.94859	0.268099
$880$	0	0
$881$	6.49000	0.218654	0.109327	−	0.994006i	$-0.465131\pi$
$881$	6.49000	0.218654	0.109327	+	0.994006i	$0.465131\pi$
$882$	0	0
$883$	22.0454	0.741887	0.370944	−	0.928655i	$-0.379034\pi$
$883$	22.0454	0.741887	0.370944	+	0.928655i	$0.379034\pi$
$884$	0	0
$885$	5.51000	0.185217
$886$	0	0
$887$	−27.7445	−0.931571	−0.465785	−	0.884898i	$-0.654228\pi$
$887$	−27.7445	−0.931571	−0.465785	+	0.884898i	$0.654228\pi$
$888$	0	0
$889$	−30.0000	−1.00617
$890$	0	0
$891$	−45.8912	−1.53741
$892$	0	0
$893$	30.9800	1.03671
$894$	0	0
$895$	−14.6969	−0.491264
$896$	0	0
$897$	49.9600	1.66812
$898$	0	0
$899$	7.34847	0.245085
$900$	0	0
$901$	−41.9600	−1.39789
$902$	0	0
$903$	−46.4913	−1.54713
$904$	0	0
$905$	−10.4900	−0.348699
$906$	0	0
$907$	−2.04941	−0.0680495	−0.0340248	−	0.999421i	$-0.510833\pi$
$907$	−2.04941	−0.0680495	−0.0340248	+	0.999421i	$0.510833\pi$
$908$	0	0
$909$	−37.4700	−1.24280
$910$	0	0
$911$	−32.6435	−1.08153	−0.540764	−	0.841174i	$-0.681865\pi$
$911$	−32.6435	−1.08153	−0.540764	+	0.841174i	$0.681865\pi$
$912$	0	0
$913$	89.4700	2.96102
$914$	0	0
$915$	35.4931	1.17337
$916$	0	0
$917$	−6.97999	−0.230500
$918$	0	0
$919$	−2.24945	−0.0742025	−0.0371012	−	0.999312i	$-0.511812\pi$
$919$	−2.24945	−0.0742025	−0.0371012	+	0.999312i	$0.511812\pi$
$920$	0	0
$921$	−30.0000	−0.988534
$922$	0	0
$923$	−10.5981	−0.348841
$924$	0	0
$925$	−9.24500	−0.303974
$926$	0	0
$927$	−29.9940	−0.985132
$928$	0	0
$929$	25.4700	0.835643	0.417822	−	0.908529i	$-0.362794\pi$
$929$	25.4700	0.835643	0.417822	+	0.908529i	$0.362794\pi$
$930$	0	0
$931$	−2.44949	−0.0802788
$932$	0	0
$933$	−6.00000	−0.196431
$934$	0	0
$935$	26.7443	0.874634
$936$	0	0
$937$	−34.4900	−1.12674	−0.563370	−	0.826205i	$-0.690495\pi$
$937$	−34.4900	−1.12674	−0.563370	+	0.826205i	$0.690495\pi$
$938$	0	0
$939$	−68.5857	−2.23821
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	−40.7922	−1.32838
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.8456	0.742381	0.371191	−	0.928557i	$-0.378950\pi$
$947$	22.8456	0.742381	0.371191	+	0.928557i	$0.378950\pi$
$948$	0	0
$949$	−3.02001	−0.0980336
$950$	0	0
$951$	36.1422	1.17199
$952$	0	0
$953$	1.02001	0.0330413	0.0165207	−	0.999864i	$-0.494741\pi$
$953$	1.02001	0.0330413	0.0165207	+	0.999864i	$0.494741\pi$
$954$	0	0
$955$	25.0950	0.812056
$956$	0	0
$957$	12.4900	0.403744
$958$	0	0
$959$	−12.8476	−0.414870
$960$	0	0
$961$	23.0000	0.741935
$962$	0	0
$963$	−22.0454	−0.710403
$964$	0	0
$965$	−19.7350	−0.635292
$966$	0	0
$967$	33.0436	1.06261	0.531305	−	0.847180i	$-0.321702\pi$
$967$	33.0436	1.06261	0.531305	+	0.847180i	$0.321702\pi$
$968$	0	0
$969$	−31.4700	−1.01096
$970$	0	0
$971$	−17.9466	−0.575933	−0.287967	−	0.957640i	$-0.592979\pi$
$971$	−17.9466	−0.575933	−0.287967	+	0.957640i	$0.592979\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	9.79796	0.313786
$976$	0	0
$977$	−6.49000	−0.207633	−0.103817	−	0.994596i	$-0.533106\pi$
$977$	−6.49000	−0.207633	−0.103817	+	0.994596i	$0.533106\pi$
$978$	0	0
$979$	−30.5941	−0.977792
$980$	0	0
$981$	48.0000	1.53252
$982$	0	0
$983$	37.1424	1.18466	0.592330	−	0.805696i	$-0.298208\pi$
$983$	37.1424	1.18466	0.592330	+	0.805696i	$0.298208\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	75.8852	2.41545
$988$	0	0
$989$	−39.5100	−1.25635
$990$	0	0
$991$	7.54851	0.239786	0.119893	−	0.992787i	$-0.461745\pi$
$991$	7.54851	0.239786	0.119893	+	0.992787i	$0.461745\pi$
$992$	0	0
$993$	−17.0200	−0.540114
$994$	0	0
$995$	7.94859	0.251987
$996$	0	0
$997$	24.2250	0.767213	0.383607	−	0.923497i	$-0.374682\pi$
$997$	24.2250	0.767213	0.383607	+	0.923497i	$0.374682\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4640.2.a.o.1.4	yes	4
4.3	odd	2	inner	4640.2.a.o.1.1	✓	4
8.3	odd	2		9280.2.a.cd.1.4		4
8.5	even	2		9280.2.a.cd.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4640.2.a.o.1.1	✓	4	4.3	odd	2	inner
4640.2.a.o.1.4	yes	4	1.1	even	1	trivial
9280.2.a.cd.1.1		4	8.5	even	2
9280.2.a.cd.1.4		4	8.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4640 = 2^{5} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4640.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44949 q^{3} -1.00000 q^{5} +2.44949 q^{7} +3.00000 q^{9} +5.09902 q^{11} +4.00000 q^{13} -2.44949 q^{15} -5.24500 q^{17} +2.44949 q^{19} +6.00000 q^{21} +5.09902 q^{23} +1.00000 q^{25} +1.00000 q^{29} +7.34847 q^{31} +12.4900 q^{33} -2.44949 q^{35} -9.24500 q^{37} +9.79796 q^{39} -8.00000 q^{41} -7.74855 q^{43} -3.00000 q^{45} +12.6475 q^{47} -1.00000 q^{49} -12.8476 q^{51} +8.00000 q^{53} -5.09902 q^{55} +6.00000 q^{57} -2.24945 q^{59} -14.4900 q^{61} +7.34847 q^{63} -4.00000 q^{65} -0.200040 q^{67} +12.4900 q^{69} -2.64953 q^{71} -0.755002 q^{73} +2.44949 q^{75} +12.4900 q^{77} +14.8970 q^{79} -9.00000 q^{81} +17.5465 q^{83} +5.24500 q^{85} +2.44949 q^{87} -6.00000 q^{89} +9.79796 q^{91} +18.0000 q^{93} -2.44949 q^{95} +11.2450 q^{97} +15.2971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 12 q^{9} + 16 q^{13} + 4 q^{17} + 24 q^{21} + 4 q^{25} + 4 q^{29} - 12 q^{37} - 32 q^{41} - 12 q^{45} - 4 q^{49} + 32 q^{53} + 24 q^{57} - 8 q^{61} - 16 q^{65} - 28 q^{73} - 36 q^{81} - 4 q^{85}+ \cdots + 20 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 + 12 * q^9 + 16 * q^13 + 4 * q^17 + 24 * q^21 + 4 * q^25 + 4 * q^29 - 12 * q^37 - 32 * q^41 - 12 * q^45 - 4 * q^49 + 32 * q^53 + 24 * q^57 - 8 * q^61 - 16 * q^65 - 28 * q^73 - 36 * q^81 - 4 * q^85 - 24 * q^89 + 72 * q^93 + 20 * q^97

Introduction

L-functions

Modular forms

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Database

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