LMFDB - Embedded newform 6384.2.a.bx.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6384,2,Mod(1,6384)] 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("6384.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,4,0,3,0,3,0,0,0,6,0,4,0,8,0,-3,0,3,0,4,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	yes
Analytic conductor:	$50.9764966504$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	6384.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.70928 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.07838 q^{11} +0.921622 q^{13} +3.70928 q^{15} +0.290725 q^{17} -1.00000 q^{19} +1.00000 q^{21} +7.60197 q^{23} +8.75872 q^{25} +1.00000 q^{27} +5.36910 q^{29} -8.49693 q^{31} +1.07838 q^{33} +3.70928 q^{35} -10.6803 q^{37} +0.921622 q^{39} -3.75872 q^{41} +8.49693 q^{43} +3.70928 q^{45} +6.20620 q^{47} +1.00000 q^{49} +0.290725 q^{51} +4.78765 q^{53} +4.00000 q^{55} -1.00000 q^{57} -4.00000 q^{59} -2.68035 q^{61} +1.00000 q^{63} +3.41855 q^{65} -1.26180 q^{67} +7.60197 q^{69} +3.86603 q^{71} +1.41855 q^{73} +8.75872 q^{75} +1.07838 q^{77} +10.5236 q^{79} +1.00000 q^{81} -8.72979 q^{83} +1.07838 q^{85} +5.36910 q^{87} -3.75872 q^{89} +0.921622 q^{91} -8.49693 q^{93} -3.70928 q^{95} -13.5174 q^{97} +1.07838 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9} + 6 q^{13} + 4 q^{15} + 8 q^{17} - 3 q^{19} + 3 q^{21} + 4 q^{23} + q^{25} + 3 q^{27} + 20 q^{29} - 8 q^{31} + 4 q^{35} - 10 q^{37} + 6 q^{39} + 14 q^{41} + 8 q^{43}+ \cdots + 10 q^{97}+O(q^{100}) $ 3 * q + 3 * q^3 + 4 * q^5 + 3 * q^7 + 3 * q^9 + 6 * q^13 + 4 * q^15 + 8 * q^17 - 3 * q^19 + 3 * q^21 + 4 * q^23 + q^25 + 3 * q^27 + 20 * q^29 - 8 * q^31 + 4 * q^35 - 10 * q^37 + 6 * q^39 + 14 * q^41 + 8 * q^43 + 4 * q^45 - 6 * q^47 + 3 * q^49 + 8 * q^51 + 4 * q^53 + 12 * q^55 - 3 * q^57 - 12 * q^59 + 14 * q^61 + 3 * q^63 - 4 * q^65 + 4 * q^67 + 4 * q^69 - 2 * q^71 - 10 * q^73 + q^75 + 16 * q^79 + 3 * q^81 + 14 * q^83 + 20 * q^87 + 14 * q^89 + 6 * q^91 - 8 * q^93 - 4 * q^95 + 10 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.70928	1.65884	0.829419	−	0.558627i	$-0.188672\pi$
$5$	3.70928	1.65884	0.829419	+	0.558627i	$0.188672\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.07838	0.325143	0.162572	−	0.986697i	$-0.448021\pi$
$11$	1.07838	0.325143	0.162572	+	0.986697i	$0.448021\pi$
$12$	0	0
$13$	0.921622	0.255612	0.127806	−	0.991799i	$-0.459207\pi$
$13$	0.921622	0.255612	0.127806	+	0.991799i	$0.459207\pi$
$14$	0	0
$15$	3.70928	0.957731
$16$	0	0
$17$	0.290725	0.0705111	0.0352555	−	0.999378i	$-0.488775\pi$
$17$	0.290725	0.0705111	0.0352555	+	0.999378i	$0.488775\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	7.60197	1.58512	0.792560	−	0.609794i	$-0.208748\pi$
$23$	7.60197	1.58512	0.792560	+	0.609794i	$0.208748\pi$
$24$	0	0
$25$	8.75872	1.75174
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.36910	0.997017	0.498509	−	0.866885i	$-0.333881\pi$
$29$	5.36910	0.997017	0.498509	+	0.866885i	$0.333881\pi$
$30$	0	0
$31$	−8.49693	−1.52609	−0.763047	−	0.646343i	$-0.776297\pi$
$31$	−8.49693	−1.52609	−0.763047	+	0.646343i	$0.776297\pi$
$32$	0	0
$33$	1.07838	0.187721
$34$	0	0
$35$	3.70928	0.626982
$36$	0	0
$37$	−10.6803	−1.75584	−0.877919	−	0.478809i	$-0.841069\pi$
$37$	−10.6803	−1.75584	−0.877919	+	0.478809i	$0.841069\pi$
$38$	0	0
$39$	0.921622	0.147578
$40$	0	0
$41$	−3.75872	−0.587014	−0.293507	−	0.955957i	$-0.594822\pi$
$41$	−3.75872	−0.587014	−0.293507	+	0.955957i	$0.594822\pi$
$42$	0	0
$43$	8.49693	1.29577	0.647885	−	0.761738i	$-0.275654\pi$
$43$	8.49693	1.29577	0.647885	+	0.761738i	$0.275654\pi$
$44$	0	0
$45$	3.70928	0.552946
$46$	0	0
$47$	6.20620	0.905268	0.452634	−	0.891696i	$-0.350484\pi$
$47$	6.20620	0.905268	0.452634	+	0.891696i	$0.350484\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.290725	0.0407096
$52$	0	0
$53$	4.78765	0.657635	0.328817	−	0.944394i	$-0.393350\pi$
$53$	4.78765	0.657635	0.328817	+	0.944394i	$0.393350\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−2.68035	−0.343183	−0.171592	−	0.985168i	$-0.554891\pi$
$61$	−2.68035	−0.343183	−0.171592	+	0.985168i	$0.554891\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	3.41855	0.424019
$66$	0	0
$67$	−1.26180	−0.154153	−0.0770764	−	0.997025i	$-0.524559\pi$
$67$	−1.26180	−0.154153	−0.0770764	+	0.997025i	$0.524559\pi$
$68$	0	0
$69$	7.60197	0.915169
$70$	0	0
$71$	3.86603	0.458813	0.229407	−	0.973331i	$-0.426321\pi$
$71$	3.86603	0.458813	0.229407	+	0.973331i	$0.426321\pi$
$72$	0	0
$73$	1.41855	0.166029	0.0830144	−	0.996548i	$-0.473545\pi$
$73$	1.41855	0.166029	0.0830144	+	0.996548i	$0.473545\pi$
$74$	0	0
$75$	8.75872	1.01137
$76$	0	0
$77$	1.07838	0.122893
$78$	0	0
$79$	10.5236	1.18400	0.591998	−	0.805939i	$-0.298339\pi$
$79$	10.5236	1.18400	0.591998	+	0.805939i	$0.298339\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.72979	−0.958219	−0.479110	−	0.877755i	$-0.659040\pi$
$83$	−8.72979	−0.958219	−0.479110	+	0.877755i	$0.659040\pi$
$84$	0	0
$85$	1.07838	0.116966
$86$	0	0
$87$	5.36910	0.575628
$88$	0	0
$89$	−3.75872	−0.398424	−0.199212	−	0.979956i	$-0.563838\pi$
$89$	−3.75872	−0.398424	−0.199212	+	0.979956i	$0.563838\pi$
$90$	0	0
$91$	0.921622	0.0966123
$92$	0	0
$93$	−8.49693	−0.881090
$94$	0	0
$95$	−3.70928	−0.380564
$96$	0	0
$97$	−13.5174	−1.37249	−0.686244	−	0.727371i	$-0.740742\pi$
$97$	−13.5174	−1.37249	−0.686244	+	0.727371i	$0.740742\pi$
$98$	0	0
$99$	1.07838	0.108381
$100$	0	0
$101$	−2.44748	−0.243533	−0.121767	−	0.992559i	$-0.538856\pi$
$101$	−2.44748	−0.243533	−0.121767	+	0.992559i	$0.538856\pi$
$102$	0	0
$103$	−19.5174	−1.92311	−0.961556	−	0.274610i	$-0.911451\pi$
$103$	−19.5174	−1.92311	−0.961556	+	0.274610i	$0.911451\pi$
$104$	0	0
$105$	3.70928	0.361988
$106$	0	0
$107$	−15.6514	−1.51308	−0.756540	−	0.653948i	$-0.773112\pi$
$107$	−15.6514	−1.51308	−0.756540	+	0.653948i	$0.773112\pi$
$108$	0	0
$109$	14.0989	1.35043	0.675215	−	0.737621i	$-0.264051\pi$
$109$	14.0989	1.35043	0.675215	+	0.737621i	$0.264051\pi$
$110$	0	0
$111$	−10.6803	−1.01373
$112$	0	0
$113$	18.7298	1.76195	0.880975	−	0.473162i	$-0.156887\pi$
$113$	18.7298	1.76195	0.880975	+	0.473162i	$0.156887\pi$
$114$	0	0
$115$	28.1978	2.62946
$116$	0	0
$117$	0.921622	0.0852040
$118$	0	0
$119$	0.290725	0.0266507
$120$	0	0
$121$	−9.83710	−0.894282
$122$	0	0
$123$	−3.75872	−0.338913
$124$	0	0
$125$	13.9421	1.24702
$126$	0	0
$127$	15.7854	1.40073	0.700363	−	0.713787i	$-0.253021\pi$
$127$	15.7854	1.40073	0.700363	+	0.713787i	$0.253021\pi$
$128$	0	0
$129$	8.49693	0.748113
$130$	0	0
$131$	−6.10731	−0.533598	−0.266799	−	0.963752i	$-0.585966\pi$
$131$	−6.10731	−0.533598	−0.266799	+	0.963752i	$0.585966\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	3.70928	0.319244
$136$	0	0
$137$	18.9939	1.62275	0.811377	−	0.584523i	$-0.198718\pi$
$137$	18.9939	1.62275	0.811377	+	0.584523i	$0.198718\pi$
$138$	0	0
$139$	−0.894960	−0.0759095	−0.0379548	−	0.999279i	$-0.512084\pi$
$139$	−0.894960	−0.0759095	−0.0379548	+	0.999279i	$0.512084\pi$
$140$	0	0
$141$	6.20620	0.522657
$142$	0	0
$143$	0.993857	0.0831105
$144$	0	0
$145$	19.9155	1.65389
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	20.9360	1.71514	0.857572	−	0.514364i	$-0.171972\pi$
$149$	20.9360	1.71514	0.857572	+	0.514364i	$0.171972\pi$
$150$	0	0
$151$	−13.9421	−1.13460	−0.567298	−	0.823513i	$-0.692011\pi$
$151$	−13.9421	−1.13460	−0.567298	+	0.823513i	$0.692011\pi$
$152$	0	0
$153$	0.290725	0.0235037
$154$	0	0
$155$	−31.5174	−2.53154
$156$	0	0
$157$	−2.31351	−0.184638	−0.0923191	−	0.995729i	$-0.529428\pi$
$157$	−2.31351	−0.184638	−0.0923191	+	0.995729i	$0.529428\pi$
$158$	0	0
$159$	4.78765	0.379686
$160$	0	0
$161$	7.60197	0.599119
$162$	0	0
$163$	17.6598	1.38322	0.691612	−	0.722269i	$-0.256901\pi$
$163$	17.6598	1.38322	0.691612	+	0.722269i	$0.256901\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	−3.05172	−0.236149	−0.118074	−	0.993005i	$-0.537672\pi$
$167$	−3.05172	−0.236149	−0.118074	+	0.993005i	$0.537672\pi$
$168$	0	0
$169$	−12.1506	−0.934662
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	9.60197	0.730024	0.365012	−	0.931003i	$-0.381065\pi$
$173$	9.60197	0.730024	0.365012	+	0.931003i	$0.381065\pi$
$174$	0	0
$175$	8.75872	0.662097
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	0.814315	0.0608648	0.0304324	−	0.999537i	$-0.490312\pi$
$179$	0.814315	0.0608648	0.0304324	+	0.999537i	$0.490312\pi$
$180$	0	0
$181$	20.1568	1.49824	0.749120	−	0.662434i	$-0.230477\pi$
$181$	20.1568	1.49824	0.749120	+	0.662434i	$0.230477\pi$
$182$	0	0
$183$	−2.68035	−0.198137
$184$	0	0
$185$	−39.6163	−2.91265
$186$	0	0
$187$	0.313511	0.0229262
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−5.44521	−0.394002	−0.197001	−	0.980403i	$-0.563120\pi$
$191$	−5.44521	−0.394002	−0.197001	+	0.980403i	$0.563120\pi$
$192$	0	0
$193$	1.68649	0.121396	0.0606981	−	0.998156i	$-0.480667\pi$
$193$	1.68649	0.121396	0.0606981	+	0.998156i	$0.480667\pi$
$194$	0	0
$195$	3.41855	0.244808
$196$	0	0
$197$	−10.3668	−0.738606	−0.369303	−	0.929309i	$-0.620404\pi$
$197$	−10.3668	−0.738606	−0.369303	+	0.929309i	$0.620404\pi$
$198$	0	0
$199$	2.42469	0.171882	0.0859410	−	0.996300i	$-0.472610\pi$
$199$	2.42469	0.171882	0.0859410	+	0.996300i	$0.472610\pi$
$200$	0	0
$201$	−1.26180	−0.0890002
$202$	0	0
$203$	5.36910	0.376837
$204$	0	0
$205$	−13.9421	−0.973761
$206$	0	0
$207$	7.60197	0.528373
$208$	0	0
$209$	−1.07838	−0.0745929
$210$	0	0
$211$	−19.5174	−1.34364	−0.671818	−	0.740716i	$-0.734486\pi$
$211$	−19.5174	−1.34364	−0.671818	+	0.740716i	$0.734486\pi$
$212$	0	0
$213$	3.86603	0.264896
$214$	0	0
$215$	31.5174	2.14947
$216$	0	0
$217$	−8.49693	−0.576809
$218$	0	0
$219$	1.41855	0.0958568
$220$	0	0
$221$	0.267938	0.0180235
$222$	0	0
$223$	−19.0205	−1.27371	−0.636854	−	0.770984i	$-0.719765\pi$
$223$	−19.0205	−1.27371	−0.636854	+	0.770984i	$0.719765\pi$
$224$	0	0
$225$	8.75872	0.583915
$226$	0	0
$227$	−11.7321	−0.778684	−0.389342	−	0.921093i	$-0.627298\pi$
$227$	−11.7321	−0.778684	−0.389342	+	0.921093i	$0.627298\pi$
$228$	0	0
$229$	−27.4596	−1.81458	−0.907290	−	0.420505i	$-0.861853\pi$
$229$	−27.4596	−1.81458	−0.907290	+	0.420505i	$0.861853\pi$
$230$	0	0
$231$	1.07838	0.0709520
$232$	0	0
$233$	−5.10504	−0.334442	−0.167221	−	0.985919i	$-0.553479\pi$
$233$	−5.10504	−0.334442	−0.167221	+	0.985919i	$0.553479\pi$
$234$	0	0
$235$	23.0205	1.50169
$236$	0	0
$237$	10.5236	0.683581
$238$	0	0
$239$	10.6537	0.689130	0.344565	−	0.938763i	$-0.388027\pi$
$239$	10.6537	0.689130	0.344565	+	0.938763i	$0.388027\pi$
$240$	0	0
$241$	−13.5174	−0.870735	−0.435368	−	0.900253i	$-0.643382\pi$
$241$	−13.5174	−0.870735	−0.435368	+	0.900253i	$0.643382\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.70928	0.236977
$246$	0	0
$247$	−0.921622	−0.0586414
$248$	0	0
$249$	−8.72979	−0.553228
$250$	0	0
$251$	18.2062	1.14917	0.574583	−	0.818447i	$-0.305164\pi$
$251$	18.2062	1.14917	0.574583	+	0.818447i	$0.305164\pi$
$252$	0	0
$253$	8.19779	0.515391
$254$	0	0
$255$	1.07838	0.0675306
$256$	0	0
$257$	12.3402	0.769759	0.384879	−	0.922967i	$-0.374243\pi$
$257$	12.3402	0.769759	0.384879	+	0.922967i	$0.374243\pi$
$258$	0	0
$259$	−10.6803	−0.663644
$260$	0	0
$261$	5.36910	0.332339
$262$	0	0
$263$	−7.50307	−0.462659	−0.231330	−	0.972875i	$-0.574308\pi$
$263$	−7.50307	−0.462659	−0.231330	+	0.972875i	$0.574308\pi$
$264$	0	0
$265$	17.7587	1.09091
$266$	0	0
$267$	−3.75872	−0.230030
$268$	0	0
$269$	−4.34017	−0.264625	−0.132313	−	0.991208i	$-0.542240\pi$
$269$	−4.34017	−0.264625	−0.132313	+	0.991208i	$0.542240\pi$
$270$	0	0
$271$	−26.0410	−1.58188	−0.790940	−	0.611893i	$-0.790408\pi$
$271$	−26.0410	−1.58188	−0.790940	+	0.611893i	$0.790408\pi$
$272$	0	0
$273$	0.921622	0.0557791
$274$	0	0
$275$	9.44521	0.569568
$276$	0	0
$277$	6.59583	0.396305	0.198152	−	0.980171i	$-0.436506\pi$
$277$	6.59583	0.396305	0.198152	+	0.980171i	$0.436506\pi$
$278$	0	0
$279$	−8.49693	−0.508698
$280$	0	0
$281$	−11.6248	−0.693475	−0.346737	−	0.937962i	$-0.612710\pi$
$281$	−11.6248	−0.693475	−0.346737	+	0.937962i	$0.612710\pi$
$282$	0	0
$283$	17.9421	1.06655	0.533275	−	0.845942i	$-0.320961\pi$
$283$	17.9421	1.06655	0.533275	+	0.845942i	$0.320961\pi$
$284$	0	0
$285$	−3.70928	−0.219719
$286$	0	0
$287$	−3.75872	−0.221870
$288$	0	0
$289$	−16.9155	−0.995028
$290$	0	0
$291$	−13.5174	−0.792407
$292$	0	0
$293$	−23.7587	−1.38800	−0.694000	−	0.719975i	$-0.744153\pi$
$293$	−23.7587	−1.38800	−0.694000	+	0.719975i	$0.744153\pi$
$294$	0	0
$295$	−14.8371	−0.863849
$296$	0	0
$297$	1.07838	0.0625738
$298$	0	0
$299$	7.00614	0.405176
$300$	0	0
$301$	8.49693	0.489755
$302$	0	0
$303$	−2.44748	−0.140604
$304$	0	0
$305$	−9.94214	−0.569285
$306$	0	0
$307$	24.0144	1.37057	0.685286	−	0.728274i	$-0.259677\pi$
$307$	24.0144	1.37057	0.685286	+	0.728274i	$0.259677\pi$
$308$	0	0
$309$	−19.5174	−1.11031
$310$	0	0
$311$	−22.9854	−1.30339	−0.651693	−	0.758483i	$-0.725941\pi$
$311$	−22.9854	−1.30339	−0.651693	+	0.758483i	$0.725941\pi$
$312$	0	0
$313$	0.523590	0.0295951	0.0147975	−	0.999891i	$-0.495290\pi$
$313$	0.523590	0.0295951	0.0147975	+	0.999891i	$0.495290\pi$
$314$	0	0
$315$	3.70928	0.208994
$316$	0	0
$317$	−17.1012	−0.960497	−0.480249	−	0.877132i	$-0.659454\pi$
$317$	−17.1012	−0.960497	−0.480249	+	0.877132i	$0.659454\pi$
$318$	0	0
$319$	5.78992	0.324173
$320$	0	0
$321$	−15.6514	−0.873577
$322$	0	0
$323$	−0.290725	−0.0161764
$324$	0	0
$325$	8.07223	0.447767
$326$	0	0
$327$	14.0989	0.779671
$328$	0	0
$329$	6.20620	0.342159
$330$	0	0
$331$	−25.1461	−1.38215	−0.691077	−	0.722781i	$-0.742863\pi$
$331$	−25.1461	−1.38215	−0.691077	+	0.722781i	$0.742863\pi$
$332$	0	0
$333$	−10.6803	−0.585279
$334$	0	0
$335$	−4.68035	−0.255715
$336$	0	0
$337$	−11.5753	−0.630547	−0.315274	−	0.949001i	$-0.602096\pi$
$337$	−11.5753	−0.630547	−0.315274	+	0.949001i	$0.602096\pi$
$338$	0	0
$339$	18.7298	1.01726
$340$	0	0
$341$	−9.16290	−0.496199
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	28.1978	1.51812
$346$	0	0
$347$	32.0144	1.71862	0.859311	−	0.511454i	$-0.170893\pi$
$347$	32.0144	1.71862	0.859311	+	0.511454i	$0.170893\pi$
$348$	0	0
$349$	−2.36683	−0.126694	−0.0633469	−	0.997992i	$-0.520177\pi$
$349$	−2.36683	−0.126694	−0.0633469	+	0.997992i	$0.520177\pi$
$350$	0	0
$351$	0.921622	0.0491926
$352$	0	0
$353$	15.4413	0.821859	0.410930	−	0.911667i	$-0.365204\pi$
$353$	15.4413	0.821859	0.410930	+	0.911667i	$0.365204\pi$
$354$	0	0
$355$	14.3402	0.761097
$356$	0	0
$357$	0.290725	0.0153868
$358$	0	0
$359$	−23.4329	−1.23674	−0.618371	−	0.785886i	$-0.712207\pi$
$359$	−23.4329	−1.23674	−0.618371	+	0.785886i	$0.712207\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−9.83710	−0.516314
$364$	0	0
$365$	5.26180	0.275415
$366$	0	0
$367$	31.3028	1.63399	0.816997	−	0.576642i	$-0.195637\pi$
$367$	31.3028	1.63399	0.816997	+	0.576642i	$0.195637\pi$
$368$	0	0
$369$	−3.75872	−0.195671
$370$	0	0
$371$	4.78765	0.248563
$372$	0	0
$373$	−23.6742	−1.22580	−0.612902	−	0.790159i	$-0.709998\pi$
$373$	−23.6742	−1.22580	−0.612902	+	0.790159i	$0.709998\pi$
$374$	0	0
$375$	13.9421	0.719969
$376$	0	0
$377$	4.94828	0.254850
$378$	0	0
$379$	7.41855	0.381065	0.190533	−	0.981681i	$-0.438978\pi$
$379$	7.41855	0.381065	0.190533	+	0.981681i	$0.438978\pi$
$380$	0	0
$381$	15.7854	0.808710
$382$	0	0
$383$	7.10504	0.363051	0.181525	−	0.983386i	$-0.441897\pi$
$383$	7.10504	0.363051	0.181525	+	0.983386i	$0.441897\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	8.49693	0.431923
$388$	0	0
$389$	16.2557	0.824194	0.412097	−	0.911140i	$-0.364796\pi$
$389$	16.2557	0.824194	0.412097	+	0.911140i	$0.364796\pi$
$390$	0	0
$391$	2.21008	0.111769
$392$	0	0
$393$	−6.10731	−0.308073
$394$	0	0
$395$	39.0349	1.96406
$396$	0	0
$397$	26.5113	1.33056	0.665282	−	0.746592i	$-0.268311\pi$
$397$	26.5113	1.33056	0.665282	+	0.746592i	$0.268311\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−6.04945	−0.302095	−0.151048	−	0.988527i	$-0.548265\pi$
$401$	−6.04945	−0.302095	−0.151048	+	0.988527i	$0.548265\pi$
$402$	0	0
$403$	−7.83096	−0.390088
$404$	0	0
$405$	3.70928	0.184315
$406$	0	0
$407$	−11.5174	−0.570899
$408$	0	0
$409$	37.1194	1.83544	0.917718	−	0.397231i	$-0.130029\pi$
$409$	37.1194	1.83544	0.917718	+	0.397231i	$0.130029\pi$
$410$	0	0
$411$	18.9939	0.936898
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−32.3812	−1.58953
$416$	0	0
$417$	−0.894960	−0.0438264
$418$	0	0
$419$	23.3523	1.14083	0.570417	−	0.821355i	$-0.306782\pi$
$419$	23.3523	1.14083	0.570417	+	0.821355i	$0.306782\pi$
$420$	0	0
$421$	7.57531	0.369198	0.184599	−	0.982814i	$-0.440901\pi$
$421$	7.57531	0.369198	0.184599	+	0.982814i	$0.440901\pi$
$422$	0	0
$423$	6.20620	0.301756
$424$	0	0
$425$	2.54638	0.123517
$426$	0	0
$427$	−2.68035	−0.129711
$428$	0	0
$429$	0.993857	0.0479839
$430$	0	0
$431$	5.90707	0.284533	0.142267	−	0.989828i	$-0.454561\pi$
$431$	5.90707	0.284533	0.142267	+	0.989828i	$0.454561\pi$
$432$	0	0
$433$	−7.47641	−0.359293	−0.179647	−	0.983731i	$-0.557495\pi$
$433$	−7.47641	−0.359293	−0.179647	+	0.983731i	$0.557495\pi$
$434$	0	0
$435$	19.9155	0.954874
$436$	0	0
$437$	−7.60197	−0.363651
$438$	0	0
$439$	9.49079	0.452970	0.226485	−	0.974015i	$-0.427276\pi$
$439$	9.49079	0.452970	0.226485	+	0.974015i	$0.427276\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	25.1773	1.19621	0.598104	−	0.801418i	$-0.295921\pi$
$443$	25.1773	1.19621	0.598104	+	0.801418i	$0.295921\pi$
$444$	0	0
$445$	−13.9421	−0.660921
$446$	0	0
$447$	20.9360	0.990239
$448$	0	0
$449$	−32.4040	−1.52924	−0.764620	−	0.644482i	$-0.777073\pi$
$449$	−32.4040	−1.52924	−0.764620	+	0.644482i	$0.777073\pi$
$450$	0	0
$451$	−4.05332	−0.190864
$452$	0	0
$453$	−13.9421	−0.655059
$454$	0	0
$455$	3.41855	0.160264
$456$	0	0
$457$	8.86830	0.414841	0.207421	−	0.978252i	$-0.433493\pi$
$457$	8.86830	0.414841	0.207421	+	0.978252i	$0.433493\pi$
$458$	0	0
$459$	0.290725	0.0135699
$460$	0	0
$461$	8.70313	0.405345	0.202673	−	0.979247i	$-0.435037\pi$
$461$	8.70313	0.405345	0.202673	+	0.979247i	$0.435037\pi$
$462$	0	0
$463$	6.21008	0.288607	0.144303	−	0.989533i	$-0.453906\pi$
$463$	6.21008	0.288607	0.144303	+	0.989533i	$0.453906\pi$
$464$	0	0
$465$	−31.5174	−1.46159
$466$	0	0
$467$	−16.4619	−0.761764	−0.380882	−	0.924624i	$-0.624380\pi$
$467$	−16.4619	−0.761764	−0.380882	+	0.924624i	$0.624380\pi$
$468$	0	0
$469$	−1.26180	−0.0582643
$470$	0	0
$471$	−2.31351	−0.106601
$472$	0	0
$473$	9.16290	0.421311
$474$	0	0
$475$	−8.75872	−0.401878
$476$	0	0
$477$	4.78765	0.219212
$478$	0	0
$479$	−16.5320	−0.755366	−0.377683	−	0.925935i	$-0.623279\pi$
$479$	−16.5320	−0.755366	−0.377683	+	0.925935i	$0.623279\pi$
$480$	0	0
$481$	−9.84324	−0.448813
$482$	0	0
$483$	7.60197	0.345902
$484$	0	0
$485$	−50.1399	−2.27674
$486$	0	0
$487$	25.6742	1.16341	0.581705	−	0.813400i	$-0.302386\pi$
$487$	25.6742	1.16341	0.581705	+	0.813400i	$0.302386\pi$
$488$	0	0
$489$	17.6598	0.798605
$490$	0	0
$491$	−27.3340	−1.23357	−0.616784	−	0.787133i	$-0.711565\pi$
$491$	−27.3340	−1.23357	−0.616784	+	0.787133i	$0.711565\pi$
$492$	0	0
$493$	1.56093	0.0703008
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	3.86603	0.173415
$498$	0	0
$499$	10.0410	0.449499	0.224749	−	0.974417i	$-0.427844\pi$
$499$	10.0410	0.449499	0.224749	+	0.974417i	$0.427844\pi$
$500$	0	0
$501$	−3.05172	−0.136341
$502$	0	0
$503$	−36.8287	−1.64211	−0.821055	−	0.570849i	$-0.806614\pi$
$503$	−36.8287	−1.64211	−0.821055	+	0.570849i	$0.806614\pi$
$504$	0	0
$505$	−9.07838	−0.403983
$506$	0	0
$507$	−12.1506	−0.539628
$508$	0	0
$509$	−14.1301	−0.626305	−0.313153	−	0.949703i	$-0.601385\pi$
$509$	−14.1301	−0.626305	−0.313153	+	0.949703i	$0.601385\pi$
$510$	0	0
$511$	1.41855	0.0627530
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−72.3956	−3.19013
$516$	0	0
$517$	6.69263	0.294342
$518$	0	0
$519$	9.60197	0.421480
$520$	0	0
$521$	28.1711	1.23420	0.617100	−	0.786885i	$-0.288307\pi$
$521$	28.1711	1.23420	0.617100	+	0.786885i	$0.288307\pi$
$522$	0	0
$523$	32.1834	1.40728	0.703641	−	0.710555i	$-0.251556\pi$
$523$	32.1834	1.40728	0.703641	+	0.710555i	$0.251556\pi$
$524$	0	0
$525$	8.75872	0.382262
$526$	0	0
$527$	−2.47027	−0.107606
$528$	0	0
$529$	34.7899	1.51261
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−3.46412	−0.150048
$534$	0	0
$535$	−58.0554	−2.50995
$536$	0	0
$537$	0.814315	0.0351403
$538$	0	0
$539$	1.07838	0.0464490
$540$	0	0
$541$	−41.0349	−1.76423	−0.882114	−	0.471036i	$-0.843880\pi$
$541$	−41.0349	−1.76423	−0.882114	+	0.471036i	$0.843880\pi$
$542$	0	0
$543$	20.1568	0.865009
$544$	0	0
$545$	52.2967	2.24014
$546$	0	0
$547$	−9.67420	−0.413639	−0.206820	−	0.978379i	$-0.566311\pi$
$547$	−9.67420	−0.413639	−0.206820	+	0.978379i	$0.566311\pi$
$548$	0	0
$549$	−2.68035	−0.114394
$550$	0	0
$551$	−5.36910	−0.228731
$552$	0	0
$553$	10.5236	0.447509
$554$	0	0
$555$	−39.6163	−1.68162
$556$	0	0
$557$	−40.1399	−1.70078	−0.850392	−	0.526150i	$-0.823635\pi$
$557$	−40.1399	−1.70078	−0.850392	+	0.526150i	$0.823635\pi$
$558$	0	0
$559$	7.83096	0.331214
$560$	0	0
$561$	0.313511	0.0132364
$562$	0	0
$563$	−44.0989	−1.85855	−0.929273	−	0.369393i	$-0.879566\pi$
$563$	−44.0989	−1.85855	−0.929273	+	0.369393i	$0.879566\pi$
$564$	0	0
$565$	69.4740	2.92279
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−13.4147	−0.562372	−0.281186	−	0.959653i	$-0.590728\pi$
$569$	−13.4147	−0.562372	−0.281186	+	0.959653i	$0.590728\pi$
$570$	0	0
$571$	27.8310	1.16469	0.582345	−	0.812942i	$-0.302135\pi$
$571$	27.8310	1.16469	0.582345	+	0.812942i	$0.302135\pi$
$572$	0	0
$573$	−5.44521	−0.227477
$574$	0	0
$575$	66.5835	2.77673
$576$	0	0
$577$	−34.1978	−1.42367	−0.711836	−	0.702345i	$-0.752136\pi$
$577$	−34.1978	−1.42367	−0.711836	+	0.702345i	$0.752136\pi$
$578$	0	0
$579$	1.68649	0.0700881
$580$	0	0
$581$	−8.72979	−0.362173
$582$	0	0
$583$	5.16290	0.213825
$584$	0	0
$585$	3.41855	0.141340
$586$	0	0
$587$	−33.1422	−1.36793	−0.683963	−	0.729517i	$-0.739745\pi$
$587$	−33.1422	−1.36793	−0.683963	+	0.729517i	$0.739745\pi$
$588$	0	0
$589$	8.49693	0.350110
$590$	0	0
$591$	−10.3668	−0.426435
$592$	0	0
$593$	29.7503	1.22170	0.610849	−	0.791747i	$-0.290828\pi$
$593$	29.7503	1.22170	0.610849	+	0.791747i	$0.290828\pi$
$594$	0	0
$595$	1.07838	0.0442092
$596$	0	0
$597$	2.42469	0.0992361
$598$	0	0
$599$	−2.19183	−0.0895557	−0.0447778	−	0.998997i	$-0.514258\pi$
$599$	−2.19183	−0.0895557	−0.0447778	+	0.998997i	$0.514258\pi$
$600$	0	0
$601$	31.9877	1.30481	0.652403	−	0.757872i	$-0.273761\pi$
$601$	31.9877	1.30481	0.652403	+	0.757872i	$0.273761\pi$
$602$	0	0
$603$	−1.26180	−0.0513843
$604$	0	0
$605$	−36.4885	−1.48347
$606$	0	0
$607$	27.5174	1.11690	0.558449	−	0.829539i	$-0.311397\pi$
$607$	27.5174	1.11690	0.558449	+	0.829539i	$0.311397\pi$
$608$	0	0
$609$	5.36910	0.217567
$610$	0	0
$611$	5.71978	0.231397
$612$	0	0
$613$	6.59583	0.266403	0.133201	−	0.991089i	$-0.457474\pi$
$613$	6.59583	0.266403	0.133201	+	0.991089i	$0.457474\pi$
$614$	0	0
$615$	−13.9421	−0.562201
$616$	0	0
$617$	−3.47641	−0.139955	−0.0699775	−	0.997549i	$-0.522293\pi$
$617$	−3.47641	−0.139955	−0.0699775	+	0.997549i	$0.522293\pi$
$618$	0	0
$619$	−26.9893	−1.08479	−0.542396	−	0.840123i	$-0.682483\pi$
$619$	−26.9893	−1.08479	−0.542396	+	0.840123i	$0.682483\pi$
$620$	0	0
$621$	7.60197	0.305056
$622$	0	0
$623$	−3.75872	−0.150590
$624$	0	0
$625$	7.92162	0.316865
$626$	0	0
$627$	−1.07838	−0.0430663
$628$	0	0
$629$	−3.10504	−0.123806
$630$	0	0
$631$	10.3402	0.411636	0.205818	−	0.978590i	$-0.434015\pi$
$631$	10.3402	0.411636	0.205818	+	0.978590i	$0.434015\pi$
$632$	0	0
$633$	−19.5174	−0.775749
$634$	0	0
$635$	58.5523	2.32358
$636$	0	0
$637$	0.921622	0.0365160
$638$	0	0
$639$	3.86603	0.152938
$640$	0	0
$641$	35.1955	1.39014	0.695070	−	0.718942i	$-0.255373\pi$
$641$	35.1955	1.39014	0.695070	+	0.718942i	$0.255373\pi$
$642$	0	0
$643$	−12.4657	−0.491600	−0.245800	−	0.969321i	$-0.579051\pi$
$643$	−12.4657	−0.491600	−0.245800	+	0.969321i	$0.579051\pi$
$644$	0	0
$645$	31.5174	1.24100
$646$	0	0
$647$	11.8804	0.467067	0.233533	−	0.972349i	$-0.424971\pi$
$647$	11.8804	0.467067	0.233533	+	0.972349i	$0.424971\pi$
$648$	0	0
$649$	−4.31351	−0.169320
$650$	0	0
$651$	−8.49693	−0.333021
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	−22.6537	−0.885153
$656$	0	0
$657$	1.41855	0.0553429
$658$	0	0
$659$	−12.5464	−0.488737	−0.244369	−	0.969682i	$-0.578581\pi$
$659$	−12.5464	−0.488737	−0.244369	+	0.969682i	$0.578581\pi$
$660$	0	0
$661$	11.6430	0.452860	0.226430	−	0.974027i	$-0.427294\pi$
$661$	11.6430	0.452860	0.226430	+	0.974027i	$0.427294\pi$
$662$	0	0
$663$	0.267938	0.0104059
$664$	0	0
$665$	−3.70928	−0.143840
$666$	0	0
$667$	40.8157	1.58039
$668$	0	0
$669$	−19.0205	−0.735376
$670$	0	0
$671$	−2.89043	−0.111584
$672$	0	0
$673$	41.3484	1.59386	0.796932	−	0.604069i	$-0.206455\pi$
$673$	41.3484	1.59386	0.796932	+	0.604069i	$0.206455\pi$
$674$	0	0
$675$	8.75872	0.337123
$676$	0	0
$677$	−19.1773	−0.737043	−0.368521	−	0.929619i	$-0.620136\pi$
$677$	−19.1773	−0.737043	−0.368521	+	0.929619i	$0.620136\pi$
$678$	0	0
$679$	−13.5174	−0.518752
$680$	0	0
$681$	−11.7321	−0.449574
$682$	0	0
$683$	22.2907	0.852931	0.426465	−	0.904504i	$-0.359759\pi$
$683$	22.2907	0.852931	0.426465	+	0.904504i	$0.359759\pi$
$684$	0	0
$685$	70.4534	2.69189
$686$	0	0
$687$	−27.4596	−1.04765
$688$	0	0
$689$	4.41241	0.168099
$690$	0	0
$691$	30.0410	1.14281	0.571407	−	0.820666i	$-0.306398\pi$
$691$	30.0410	1.14281	0.571407	+	0.820666i	$0.306398\pi$
$692$	0	0
$693$	1.07838	0.0409642
$694$	0	0
$695$	−3.31965	−0.125922
$696$	0	0
$697$	−1.09275	−0.0413910
$698$	0	0
$699$	−5.10504	−0.193090
$700$	0	0
$701$	−1.20394	−0.0454721	−0.0227360	−	0.999742i	$-0.507238\pi$
$701$	−1.20394	−0.0454721	−0.0227360	+	0.999742i	$0.507238\pi$
$702$	0	0
$703$	10.6803	0.402817
$704$	0	0
$705$	23.0205	0.867003
$706$	0	0
$707$	−2.44748	−0.0920470
$708$	0	0
$709$	−33.2351	−1.24817	−0.624086	−	0.781356i	$-0.714528\pi$
$709$	−33.2351	−1.24817	−0.624086	+	0.781356i	$0.714528\pi$
$710$	0	0
$711$	10.5236	0.394665
$712$	0	0
$713$	−64.5934	−2.41904
$714$	0	0
$715$	3.68649	0.137867
$716$	0	0
$717$	10.6537	0.397869
$718$	0	0
$719$	−10.2062	−0.380627	−0.190314	−	0.981723i	$-0.560950\pi$
$719$	−10.2062	−0.380627	−0.190314	+	0.981723i	$0.560950\pi$
$720$	0	0
$721$	−19.5174	−0.726868
$722$	0	0
$723$	−13.5174	−0.502719
$724$	0	0
$725$	47.0265	1.74652
$726$	0	0
$727$	26.8371	0.995333	0.497666	−	0.867368i	$-0.334190\pi$
$727$	26.8371	0.995333	0.497666	+	0.867368i	$0.334190\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.47027	0.0913661
$732$	0	0
$733$	2.46573	0.0910739	0.0455369	−	0.998963i	$-0.485500\pi$
$733$	2.46573	0.0910739	0.0455369	+	0.998963i	$0.485500\pi$
$734$	0	0
$735$	3.70928	0.136819
$736$	0	0
$737$	−1.36069	−0.0501217
$738$	0	0
$739$	−0.863763	−0.0317741	−0.0158870	−	0.999874i	$-0.505057\pi$
$739$	−0.863763	−0.0317741	−0.0158870	+	0.999874i	$0.505057\pi$
$740$	0	0
$741$	−0.921622	−0.0338566
$742$	0	0
$743$	36.1171	1.32501	0.662505	−	0.749058i	$-0.269493\pi$
$743$	36.1171	1.32501	0.662505	+	0.749058i	$0.269493\pi$
$744$	0	0
$745$	77.6574	2.84515
$746$	0	0
$747$	−8.72979	−0.319406
$748$	0	0
$749$	−15.6514	−0.571890
$750$	0	0
$751$	33.8264	1.23434	0.617172	−	0.786828i	$-0.288278\pi$
$751$	33.8264	1.23434	0.617172	+	0.786828i	$0.288278\pi$
$752$	0	0
$753$	18.2062	0.663471
$754$	0	0
$755$	−51.7152	−1.88211
$756$	0	0
$757$	−13.6865	−0.497444	−0.248722	−	0.968575i	$-0.580011\pi$
$757$	−13.6865	−0.497444	−0.248722	+	0.968575i	$0.580011\pi$
$758$	0	0
$759$	8.19779	0.297561
$760$	0	0
$761$	−7.86150	−0.284979	−0.142490	−	0.989796i	$-0.545511\pi$
$761$	−7.86150	−0.284979	−0.142490	+	0.989796i	$0.545511\pi$
$762$	0	0
$763$	14.0989	0.510414
$764$	0	0
$765$	1.07838	0.0389888
$766$	0	0
$767$	−3.68649	−0.133111
$768$	0	0
$769$	2.31351	0.0834273	0.0417137	−	0.999130i	$-0.486718\pi$
$769$	2.31351	0.0834273	0.0417137	+	0.999130i	$0.486718\pi$
$770$	0	0
$771$	12.3402	0.444420
$772$	0	0
$773$	−32.9216	−1.18411	−0.592054	−	0.805898i	$-0.701683\pi$
$773$	−32.9216	−1.18411	−0.592054	+	0.805898i	$0.701683\pi$
$774$	0	0
$775$	−74.4222	−2.67333
$776$	0	0
$777$	−10.6803	−0.383155
$778$	0	0
$779$	3.75872	0.134670
$780$	0	0
$781$	4.16904	0.149180
$782$	0	0
$783$	5.36910	0.191876
$784$	0	0
$785$	−8.58145	−0.306285
$786$	0	0
$787$	−24.8104	−0.884397	−0.442198	−	0.896917i	$-0.645801\pi$
$787$	−24.8104	−0.884397	−0.442198	+	0.896917i	$0.645801\pi$
$788$	0	0
$789$	−7.50307	−0.267116
$790$	0	0
$791$	18.7298	0.665955
$792$	0	0
$793$	−2.47027	−0.0877217
$794$	0	0
$795$	17.7587	0.629837
$796$	0	0
$797$	2.59583	0.0919488	0.0459744	−	0.998943i	$-0.485361\pi$
$797$	2.59583	0.0919488	0.0459744	+	0.998943i	$0.485361\pi$
$798$	0	0
$799$	1.80430	0.0638314
$800$	0	0
$801$	−3.75872	−0.132808
$802$	0	0
$803$	1.52973	0.0539831
$804$	0	0
$805$	28.1978	0.993842
$806$	0	0
$807$	−4.34017	−0.152781
$808$	0	0
$809$	38.6803	1.35993	0.679964	−	0.733245i	$-0.261995\pi$
$809$	38.6803	1.35993	0.679964	+	0.733245i	$0.261995\pi$
$810$	0	0
$811$	44.9939	1.57995	0.789974	−	0.613140i	$-0.210094\pi$
$811$	44.9939	1.57995	0.789974	+	0.613140i	$0.210094\pi$
$812$	0	0
$813$	−26.0410	−0.913299
$814$	0	0
$815$	65.5052	2.29455
$816$	0	0
$817$	−8.49693	−0.297270
$818$	0	0
$819$	0.921622	0.0322041
$820$	0	0
$821$	3.77310	0.131682	0.0658410	−	0.997830i	$-0.479027\pi$
$821$	3.77310	0.131682	0.0658410	+	0.997830i	$0.479027\pi$
$822$	0	0
$823$	14.6537	0.510795	0.255398	−	0.966836i	$-0.417794\pi$
$823$	14.6537	0.510795	0.255398	+	0.966836i	$0.417794\pi$
$824$	0	0
$825$	9.44521	0.328840
$826$	0	0
$827$	15.4368	0.536790	0.268395	−	0.963309i	$-0.413507\pi$
$827$	15.4368	0.536790	0.268395	+	0.963309i	$0.413507\pi$
$828$	0	0
$829$	8.57691	0.297889	0.148944	−	0.988846i	$-0.452412\pi$
$829$	8.57691	0.297889	0.148944	+	0.988846i	$0.452412\pi$
$830$	0	0
$831$	6.59583	0.228807
$832$	0	0
$833$	0.290725	0.0100730
$834$	0	0
$835$	−11.3197	−0.391733
$836$	0	0
$837$	−8.49693	−0.293697
$838$	0	0
$839$	−31.0349	−1.07144	−0.535722	−	0.844395i	$-0.679960\pi$
$839$	−31.0349	−1.07144	−0.535722	+	0.844395i	$0.679960\pi$
$840$	0	0
$841$	−0.172740	−0.00595654
$842$	0	0
$843$	−11.6248	−0.400378
$844$	0	0
$845$	−45.0700	−1.55045
$846$	0	0
$847$	−9.83710	−0.338007
$848$	0	0
$849$	17.9421	0.615773
$850$	0	0
$851$	−81.1917	−2.78321
$852$	0	0
$853$	11.4140	0.390808	0.195404	−	0.980723i	$-0.437398\pi$
$853$	11.4140	0.390808	0.195404	+	0.980723i	$0.437398\pi$
$854$	0	0
$855$	−3.70928	−0.126855
$856$	0	0
$857$	22.7936	0.778615	0.389308	−	0.921108i	$-0.372714\pi$
$857$	22.7936	0.778615	0.389308	+	0.921108i	$0.372714\pi$
$858$	0	0
$859$	30.7838	1.05033	0.525164	−	0.851001i	$-0.324004\pi$
$859$	30.7838	1.05033	0.525164	+	0.851001i	$0.324004\pi$
$860$	0	0
$861$	−3.75872	−0.128097
$862$	0	0
$863$	−10.0228	−0.341180	−0.170590	−	0.985342i	$-0.554567\pi$
$863$	−10.0228	−0.341180	−0.170590	+	0.985342i	$0.554567\pi$
$864$	0	0
$865$	35.6163	1.21099
$866$	0	0
$867$	−16.9155	−0.574480
$868$	0	0
$869$	11.3484	0.384968
$870$	0	0
$871$	−1.16290	−0.0394033
$872$	0	0
$873$	−13.5174	−0.457496
$874$	0	0
$875$	13.9421	0.471330
$876$	0	0
$877$	−32.8371	−1.10883	−0.554415	−	0.832240i	$-0.687058\pi$
$877$	−32.8371	−1.10883	−0.554415	+	0.832240i	$0.687058\pi$
$878$	0	0
$879$	−23.7587	−0.801362
$880$	0	0
$881$	−45.0700	−1.51845	−0.759223	−	0.650831i	$-0.774421\pi$
$881$	−45.0700	−1.51845	−0.759223	+	0.650831i	$0.774421\pi$
$882$	0	0
$883$	−54.5523	−1.83583	−0.917916	−	0.396774i	$-0.870130\pi$
$883$	−54.5523	−1.83583	−0.917916	+	0.396774i	$0.870130\pi$
$884$	0	0
$885$	−14.8371	−0.498744
$886$	0	0
$887$	−16.4657	−0.552865	−0.276433	−	0.961033i	$-0.589152\pi$
$887$	−16.4657	−0.552865	−0.276433	+	0.961033i	$0.589152\pi$
$888$	0	0
$889$	15.7854	0.529425
$890$	0	0
$891$	1.07838	0.0361270
$892$	0	0
$893$	−6.20620	−0.207683
$894$	0	0
$895$	3.02052	0.100965
$896$	0	0
$897$	7.00614	0.233928
$898$	0	0
$899$	−45.6209	−1.52154
$900$	0	0
$901$	1.39189	0.0463705
$902$	0	0
$903$	8.49693	0.282760
$904$	0	0
$905$	74.7670	2.48534
$906$	0	0
$907$	23.5708	0.782655	0.391327	−	0.920252i	$-0.372016\pi$
$907$	23.5708	0.782655	0.391327	+	0.920252i	$0.372016\pi$
$908$	0	0
$909$	−2.44748	−0.0811778
$910$	0	0
$911$	0.616522	0.0204263	0.0102131	−	0.999948i	$-0.496749\pi$
$911$	0.616522	0.0204263	0.0102131	+	0.999948i	$0.496749\pi$
$912$	0	0
$913$	−9.41402	−0.311558
$914$	0	0
$915$	−9.94214	−0.328677
$916$	0	0
$917$	−6.10731	−0.201681
$918$	0	0
$919$	−53.6742	−1.77055	−0.885274	−	0.465069i	$-0.846029\pi$
$919$	−53.6742	−1.77055	−0.885274	+	0.465069i	$0.846029\pi$
$920$	0	0
$921$	24.0144	0.791301
$922$	0	0
$923$	3.56302	0.117278
$924$	0	0
$925$	−93.5462	−3.07578
$926$	0	0
$927$	−19.5174	−0.641037
$928$	0	0
$929$	−50.2616	−1.64903	−0.824515	−	0.565840i	$-0.808552\pi$
$929$	−50.2616	−1.64903	−0.824515	+	0.565840i	$0.808552\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−22.9854	−0.752510
$934$	0	0
$935$	1.16290	0.0380308
$936$	0	0
$937$	8.63931	0.282234	0.141117	−	0.989993i	$-0.454931\pi$
$937$	8.63931	0.282234	0.141117	+	0.989993i	$0.454931\pi$
$938$	0	0
$939$	0.523590	0.0170867
$940$	0	0
$941$	30.2122	0.984889	0.492444	−	0.870344i	$-0.336104\pi$
$941$	30.2122	0.984889	0.492444	+	0.870344i	$0.336104\pi$
$942$	0	0
$943$	−28.5737	−0.930488
$944$	0	0
$945$	3.70928	0.120663
$946$	0	0
$947$	−27.1727	−0.882995	−0.441498	−	0.897262i	$-0.645553\pi$
$947$	−27.1727	−0.882995	−0.441498	+	0.897262i	$0.645553\pi$
$948$	0	0
$949$	1.30737	0.0424390
$950$	0	0
$951$	−17.1012	−0.554543
$952$	0	0
$953$	44.5029	1.44159	0.720795	−	0.693148i	$-0.243777\pi$
$953$	44.5029	1.44159	0.720795	+	0.693148i	$0.243777\pi$
$954$	0	0
$955$	−20.1978	−0.653585
$956$	0	0
$957$	5.78992	0.187162
$958$	0	0
$959$	18.9939	0.613344
$960$	0	0
$961$	41.1978	1.32896
$962$	0	0
$963$	−15.6514	−0.504360
$964$	0	0
$965$	6.25565	0.201377
$966$	0	0
$967$	18.6004	0.598147	0.299074	−	0.954230i	$-0.403322\pi$
$967$	18.6004	0.598147	0.299074	+	0.954230i	$0.403322\pi$
$968$	0	0
$969$	−0.290725	−0.00933942
$970$	0	0
$971$	−16.0533	−0.515176	−0.257588	−	0.966255i	$-0.582928\pi$
$971$	−16.0533	−0.515176	−0.257588	+	0.966255i	$0.582928\pi$
$972$	0	0
$973$	−0.894960	−0.0286911
$974$	0	0
$975$	8.07223	0.258518
$976$	0	0
$977$	−15.8394	−0.506746	−0.253373	−	0.967369i	$-0.581540\pi$
$977$	−15.8394	−0.506746	−0.253373	+	0.967369i	$0.581540\pi$
$978$	0	0
$979$	−4.05332	−0.129545
$980$	0	0
$981$	14.0989	0.450143
$982$	0	0
$983$	23.0349	0.734699	0.367350	−	0.930083i	$-0.380265\pi$
$983$	23.0349	0.734699	0.367350	+	0.930083i	$0.380265\pi$
$984$	0	0
$985$	−38.4534	−1.22523
$986$	0	0
$987$	6.20620	0.197546
$988$	0	0
$989$	64.5934	2.05395
$990$	0	0
$991$	42.8371	1.36077	0.680383	−	0.732857i	$-0.261814\pi$
$991$	42.8371	1.36077	0.680383	+	0.732857i	$0.261814\pi$
$992$	0	0
$993$	−25.1461	−0.797987
$994$	0	0
$995$	8.99386	0.285124
$996$	0	0
$997$	−23.7899	−0.753434	−0.376717	−	0.926328i	$-0.622947\pi$
$997$	−23.7899	−0.753434	−0.376717	+	0.926328i	$0.622947\pi$
$998$	0	0
$999$	−10.6803	−0.337911

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.bx.1.3		3
4.3	odd	2		399.2.a.d.1.3	✓	3
12.11	even	2		1197.2.a.l.1.1		3
20.19	odd	2		9975.2.a.z.1.1		3
28.27	even	2		2793.2.a.x.1.3		3
76.75	even	2		7581.2.a.n.1.1		3
84.83	odd	2		8379.2.a.bp.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.d.1.3	✓	3	4.3	odd	2
1197.2.a.l.1.1		3	12.11	even	2
2793.2.a.x.1.3		3	28.27	even	2
6384.2.a.bx.1.3		3	1.1	even	1	trivial
7581.2.a.n.1.1		3	76.75	even	2
8379.2.a.bp.1.1		3	84.83	odd	2
9975.2.a.z.1.1		3	20.19	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:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.70928 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.07838 q^{11} +0.921622 q^{13} +3.70928 q^{15} +0.290725 q^{17} -1.00000 q^{19} +1.00000 q^{21} +7.60197 q^{23} +8.75872 q^{25} +1.00000 q^{27} +5.36910 q^{29} -8.49693 q^{31} +1.07838 q^{33} +3.70928 q^{35} -10.6803 q^{37} +0.921622 q^{39} -3.75872 q^{41} +8.49693 q^{43} +3.70928 q^{45} +6.20620 q^{47} +1.00000 q^{49} +0.290725 q^{51} +4.78765 q^{53} +4.00000 q^{55} -1.00000 q^{57} -4.00000 q^{59} -2.68035 q^{61} +1.00000 q^{63} +3.41855 q^{65} -1.26180 q^{67} +7.60197 q^{69} +3.86603 q^{71} +1.41855 q^{73} +8.75872 q^{75} +1.07838 q^{77} +10.5236 q^{79} +1.00000 q^{81} -8.72979 q^{83} +1.07838 q^{85} +5.36910 q^{87} -3.75872 q^{89} +0.921622 q^{91} -8.49693 q^{93} -3.70928 q^{95} -13.5174 q^{97} +1.07838 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9} + 6 q^{13} + 4 q^{15} + 8 q^{17} - 3 q^{19} + 3 q^{21} + 4 q^{23} + q^{25} + 3 q^{27} + 20 q^{29} - 8 q^{31} + 4 q^{35} - 10 q^{37} + 6 q^{39} + 14 q^{41} + 8 q^{43}+ \cdots + 10 q^{97}+O(q^{100}) \) 3 * q + 3 * q^3 + 4 * q^5 + 3 * q^7 + 3 * q^9 + 6 * q^13 + 4 * q^15 + 8 * q^17 - 3 * q^19 + 3 * q^21 + 4 * q^23 + q^25 + 3 * q^27 + 20 * q^29 - 8 * q^31 + 4 * q^35 - 10 * q^37 + 6 * q^39 + 14 * q^41 + 8 * q^43 + 4 * q^45 - 6 * q^47 + 3 * q^49 + 8 * q^51 + 4 * q^53 + 12 * q^55 - 3 * q^57 - 12 * q^59 + 14 * q^61 + 3 * q^63 - 4 * q^65 + 4 * q^67 + 4 * q^69 - 2 * q^71 - 10 * q^73 + q^75 + 16 * q^79 + 3 * q^81 + 14 * q^83 + 20 * q^87 + 14 * q^89 + 6 * q^91 - 8 * q^93 - 4 * q^95 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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