LMFDB - Embedded newform 8624.2.a.db.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8624,2,Mod(1,8624)] 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("8624.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-1,0,-4,0,0,0,0,0,5,0,-1,0,-7,0,-9,0,1,0,0,0,8,0,5,0,-4, 0,9,0,3,0,-1,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

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

Self dual:	yes
Analytic conductor:	$68.8629867032$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.559701.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 4x^{2} + 10x - 3 $ x^5 - x^4 - 7x^3 + 4x^2 + 10*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.33366$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.33366 q^{3} -1.11230 q^{5} -1.22136 q^{9} +1.00000 q^{11} +0.380391 q^{13} -1.48342 q^{15} +3.35852 q^{17} +4.40525 q^{19} -6.54241 q^{23} -3.76279 q^{25} -5.62984 q^{27} +3.06556 q^{29} -2.07159 q^{31} +1.33366 q^{33} +4.87833 q^{37} +0.507311 q^{39} -6.56428 q^{41} -12.4155 q^{43} +1.35852 q^{45} +7.35852 q^{47} +4.47911 q^{51} +0.770347 q^{53} -1.11230 q^{55} +5.87509 q^{57} +6.72856 q^{59} -11.9477 q^{61} -0.423108 q^{65} -7.51852 q^{67} -8.72532 q^{69} +4.10572 q^{71} +5.36875 q^{73} -5.01828 q^{75} +10.6047 q^{79} -3.84420 q^{81} -16.3001 q^{83} -3.73567 q^{85} +4.08841 q^{87} -8.59572 q^{89} -2.76279 q^{93} -4.89995 q^{95} -4.51152 q^{97} -1.22136 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - 4 q^{5} + 5 q^{11} - q^{13} - 7 q^{15} - 9 q^{17} + q^{19} + 8 q^{23} + 5 q^{25} - 4 q^{27} + 9 q^{29} + 3 q^{31} - q^{33} + 2 q^{37} + 7 q^{39} - 15 q^{41} - 14 q^{43} - 19 q^{45} + 11 q^{47}+ \cdots - 7 q^{97}+O(q^{100}) $ 5 * q - q^3 - 4 * q^5 + 5 * q^11 - q^13 - 7 * q^15 - 9 * q^17 + q^19 + 8 * q^23 + 5 * q^25 - 4 * q^27 + 9 * q^29 + 3 * q^31 - q^33 + 2 * q^37 + 7 * q^39 - 15 * q^41 - 14 * q^43 - 19 * q^45 + 11 * q^47 + 14 * q^51 - 9 * q^53 - 4 * q^55 + 4 * q^57 + 4 * q^59 + 2 * q^61 + 7 * q^65 + 8 * q^67 - 11 * q^69 - 15 * q^71 - 26 * q^73 + 27 * q^75 + 3 * q^79 - 19 * q^81 - q^83 + 13 * q^85 + 14 * q^87 - 41 * q^89 + 10 * q^93 + 19 * q^95 - 7 * 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
$3$	1.33366	0.769987	0.384994	−	0.922919i	$-0.374204\pi$
$3$	1.33366	0.769987	0.384994	+	0.922919i	$0.374204\pi$
$4$	0	0
$5$	−1.11230	−0.497435	−0.248717	−	0.968576i	$-0.580009\pi$
$5$	−1.11230	−0.497435	−0.248717	+	0.968576i	$0.580009\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.22136	−0.407120
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.380391	0.105501	0.0527507	−	0.998608i	$-0.483201\pi$
$13$	0.380391	0.105501	0.0527507	+	0.998608i	$0.483201\pi$
$14$	0	0
$15$	−1.48342	−0.383018
$16$	0	0
$17$	3.35852	0.814560	0.407280	−	0.913303i	$-0.366477\pi$
$17$	3.35852	0.814560	0.407280	+	0.913303i	$0.366477\pi$
$18$	0	0
$19$	4.40525	1.01063	0.505317	−	0.862934i	$-0.331376\pi$
$19$	4.40525	1.01063	0.505317	+	0.862934i	$0.331376\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.54241	−1.36419	−0.682093	−	0.731266i	$-0.738930\pi$
$23$	−6.54241	−1.36419	−0.682093	+	0.731266i	$0.738930\pi$
$24$	0	0
$25$	−3.76279	−0.752559
$26$	0	0
$27$	−5.62984	−1.08346
$28$	0	0
$29$	3.06556	0.569261	0.284630	−	0.958637i	$-0.408129\pi$
$29$	3.06556	0.569261	0.284630	+	0.958637i	$0.408129\pi$
$30$	0	0
$31$	−2.07159	−0.372069	−0.186034	−	0.982543i	$-0.559564\pi$
$31$	−2.07159	−0.372069	−0.186034	+	0.982543i	$0.559564\pi$
$32$	0	0
$33$	1.33366	0.232160
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.87833	0.801992	0.400996	−	0.916080i	$-0.368664\pi$
$37$	4.87833	0.801992	0.400996	+	0.916080i	$0.368664\pi$
$38$	0	0
$39$	0.507311	0.0812348
$40$	0	0
$41$	−6.56428	−1.02517	−0.512584	−	0.858637i	$-0.671312\pi$
$41$	−6.56428	−1.02517	−0.512584	+	0.858637i	$0.671312\pi$
$42$	0	0
$43$	−12.4155	−1.89334	−0.946672	−	0.322199i	$-0.895578\pi$
$43$	−12.4155	−1.89334	−0.946672	+	0.322199i	$0.895578\pi$
$44$	0	0
$45$	1.35852	0.202515
$46$	0	0
$47$	7.35852	1.07335	0.536675	−	0.843789i	$-0.319680\pi$
$47$	7.35852	1.07335	0.536675	+	0.843789i	$0.319680\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.47911	0.627200
$52$	0	0
$53$	0.770347	0.105815	0.0529076	−	0.998599i	$-0.483151\pi$
$53$	0.770347	0.105815	0.0529076	+	0.998599i	$0.483151\pi$
$54$	0	0
$55$	−1.11230	−0.149982
$56$	0	0
$57$	5.87509	0.778175
$58$	0	0
$59$	6.72856	0.875984	0.437992	−	0.898979i	$-0.355690\pi$
$59$	6.72856	0.875984	0.437992	+	0.898979i	$0.355690\pi$
$60$	0	0
$61$	−11.9477	−1.52974	−0.764870	−	0.644184i	$-0.777197\pi$
$61$	−11.9477	−1.52974	−0.764870	+	0.644184i	$0.777197\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.423108	−0.0524801
$66$	0	0
$67$	−7.51852	−0.918533	−0.459267	−	0.888298i	$-0.651888\pi$
$67$	−7.51852	−0.918533	−0.459267	+	0.888298i	$0.651888\pi$
$68$	0	0
$69$	−8.72532	−1.05041
$70$	0	0
$71$	4.10572	0.487259	0.243629	−	0.969868i	$-0.421662\pi$
$71$	4.10572	0.487259	0.243629	+	0.969868i	$0.421662\pi$
$72$	0	0
$73$	5.36875	0.628365	0.314182	−	0.949363i	$-0.398270\pi$
$73$	5.36875	0.628365	0.314182	+	0.949363i	$0.398270\pi$
$74$	0	0
$75$	−5.01828	−0.579461
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.6047	1.19313	0.596563	−	0.802566i	$-0.296533\pi$
$79$	10.6047	1.19313	0.596563	+	0.802566i	$0.296533\pi$
$80$	0	0
$81$	−3.84420	−0.427134
$82$	0	0
$83$	−16.3001	−1.78917	−0.894586	−	0.446895i	$-0.852530\pi$
$83$	−16.3001	−1.78917	−0.894586	+	0.446895i	$0.852530\pi$
$84$	0	0
$85$	−3.73567	−0.405190
$86$	0	0
$87$	4.08841	0.438324
$88$	0	0
$89$	−8.59572	−0.911145	−0.455572	−	0.890199i	$-0.650565\pi$
$89$	−8.59572	−0.911145	−0.455572	+	0.890199i	$0.650565\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.76279	−0.286488
$94$	0	0
$95$	−4.89995	−0.502724
$96$	0	0
$97$	−4.51152	−0.458075	−0.229038	−	0.973418i	$-0.573558\pi$
$97$	−4.51152	−0.458075	−0.229038	+	0.973418i	$0.573558\pi$
$98$	0	0
$99$	−1.22136	−0.122751
$100$	0	0
$101$	−16.1836	−1.61033	−0.805166	−	0.593050i	$-0.797924\pi$
$101$	−16.1836	−1.61033	−0.805166	+	0.593050i	$0.797924\pi$
$102$	0	0
$103$	10.3178	1.01664	0.508322	−	0.861167i	$-0.330266\pi$
$103$	10.3178	1.01664	0.508322	+	0.861167i	$0.330266\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.38014	−0.326770	−0.163385	−	0.986562i	$-0.552241\pi$
$107$	−3.38014	−0.326770	−0.163385	+	0.986562i	$0.552241\pi$
$108$	0	0
$109$	7.09418	0.679500	0.339750	−	0.940516i	$-0.389658\pi$
$109$	7.09418	0.679500	0.339750	+	0.940516i	$0.389658\pi$
$110$	0	0
$111$	6.50602	0.617524
$112$	0	0
$113$	4.01529	0.377727	0.188864	−	0.982003i	$-0.439520\pi$
$113$	4.01529	0.377727	0.188864	+	0.982003i	$0.439520\pi$
$114$	0	0
$115$	7.27710	0.678593
$116$	0	0
$117$	−0.464594	−0.0429517
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−8.75450	−0.789367
$124$	0	0
$125$	9.74684	0.871784
$126$	0	0
$127$	−9.86381	−0.875272	−0.437636	−	0.899152i	$-0.644184\pi$
$127$	−9.86381	−0.875272	−0.437636	+	0.899152i	$0.644184\pi$
$128$	0	0
$129$	−16.5580	−1.45785
$130$	0	0
$131$	−5.84183	−0.510403	−0.255202	−	0.966888i	$-0.582142\pi$
$131$	−5.84183	−0.510403	−0.255202	+	0.966888i	$0.582142\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	6.26206	0.538953
$136$	0	0
$137$	3.99989	0.341734	0.170867	−	0.985294i	$-0.445343\pi$
$137$	3.99989	0.341734	0.170867	+	0.985294i	$0.445343\pi$
$138$	0	0
$139$	1.31997	0.111958	0.0559790	−	0.998432i	$-0.482172\pi$
$139$	1.31997	0.111958	0.0559790	+	0.998432i	$0.482172\pi$
$140$	0	0
$141$	9.81374	0.826466
$142$	0	0
$143$	0.380391	0.0318099
$144$	0	0
$145$	−3.40982	−0.283170
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.23257	−0.346745	−0.173373	−	0.984856i	$-0.555467\pi$
$149$	−4.23257	−0.346745	−0.173373	+	0.984856i	$0.555467\pi$
$150$	0	0
$151$	23.2290	1.89035	0.945173	−	0.326569i	$-0.105893\pi$
$151$	23.2290	1.89035	0.945173	+	0.326569i	$0.105893\pi$
$152$	0	0
$153$	−4.10195	−0.331623
$154$	0	0
$155$	2.30423	0.185080
$156$	0	0
$157$	−3.56331	−0.284383	−0.142192	−	0.989839i	$-0.545415\pi$
$157$	−3.56331	−0.284383	−0.142192	+	0.989839i	$0.545415\pi$
$158$	0	0
$159$	1.02738	0.0814764
$160$	0	0
$161$	0	0
$162$	0	0
$163$	21.9778	1.72143	0.860716	−	0.509086i	$-0.170017\pi$
$163$	21.9778	1.72143	0.860716	+	0.509086i	$0.170017\pi$
$164$	0	0
$165$	−1.48342	−0.115484
$166$	0	0
$167$	6.25548	0.484064	0.242032	−	0.970268i	$-0.422186\pi$
$167$	6.25548	0.484064	0.242032	+	0.970268i	$0.422186\pi$
$168$	0	0
$169$	−12.8553	−0.988869
$170$	0	0
$171$	−5.38039	−0.411449
$172$	0	0
$173$	−0.436799	−0.0332092	−0.0166046	−	0.999862i	$-0.505286\pi$
$173$	−0.436799	−0.0332092	−0.0166046	+	0.999862i	$0.505286\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.97359	0.674497
$178$	0	0
$179$	−8.18590	−0.611843	−0.305922	−	0.952057i	$-0.598965\pi$
$179$	−8.18590	−0.611843	−0.305922	+	0.952057i	$0.598965\pi$
$180$	0	0
$181$	−18.5587	−1.37946	−0.689728	−	0.724069i	$-0.742270\pi$
$181$	−18.5587	−1.37946	−0.689728	+	0.724069i	$0.742270\pi$
$182$	0	0
$183$	−15.9341	−1.17788
$184$	0	0
$185$	−5.42615	−0.398939
$186$	0	0
$187$	3.35852	0.245599
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.20149	−0.0869371	−0.0434685	−	0.999055i	$-0.513841\pi$
$191$	−1.20149	−0.0869371	−0.0434685	+	0.999055i	$0.513841\pi$
$192$	0	0
$193$	−20.4663	−1.47320	−0.736598	−	0.676331i	$-0.763569\pi$
$193$	−20.4663	−1.47320	−0.736598	+	0.676331i	$0.763569\pi$
$194$	0	0
$195$	−0.564281	−0.0404090
$196$	0	0
$197$	−15.4750	−1.10255	−0.551275	−	0.834324i	$-0.685858\pi$
$197$	−15.4750	−1.10255	−0.551275	+	0.834324i	$0.685858\pi$
$198$	0	0
$199$	−13.1745	−0.933917	−0.466958	−	0.884279i	$-0.654650\pi$
$199$	−13.1745	−0.933917	−0.466958	+	0.884279i	$0.654650\pi$
$200$	0	0
$201$	−10.0271	−0.707259
$202$	0	0
$203$	0	0
$204$	0	0
$205$	7.30144	0.509954
$206$	0	0
$207$	7.99063	0.555387
$208$	0	0
$209$	4.40525	0.304717
$210$	0	0
$211$	−13.2004	−0.908754	−0.454377	−	0.890809i	$-0.650138\pi$
$211$	−13.2004	−0.908754	−0.454377	+	0.890809i	$0.650138\pi$
$212$	0	0
$213$	5.47562	0.375183
$214$	0	0
$215$	13.8097	0.941815
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7.16007	0.483833
$220$	0	0
$221$	1.27755	0.0859372
$222$	0	0
$223$	−9.49742	−0.635994	−0.317997	−	0.948092i	$-0.603010\pi$
$223$	−9.49742	−0.635994	−0.317997	+	0.948092i	$0.603010\pi$
$224$	0	0
$225$	4.59572	0.306381
$226$	0	0
$227$	3.22984	0.214372	0.107186	−	0.994239i	$-0.465816\pi$
$227$	3.22984	0.214372	0.107186	+	0.994239i	$0.465816\pi$
$228$	0	0
$229$	10.8045	0.713979	0.356990	−	0.934108i	$-0.383803\pi$
$229$	10.8045	0.713979	0.356990	+	0.934108i	$0.383803\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−26.9444	−1.76519	−0.882594	−	0.470136i	$-0.844205\pi$
$233$	−26.9444	−1.76519	−0.882594	+	0.470136i	$0.844205\pi$
$234$	0	0
$235$	−8.18486	−0.533921
$236$	0	0
$237$	14.1431	0.918691
$238$	0	0
$239$	−4.82091	−0.311839	−0.155919	−	0.987770i	$-0.549834\pi$
$239$	−4.82091	−0.311839	−0.155919	+	0.987770i	$0.549834\pi$
$240$	0	0
$241$	−12.9673	−0.835299	−0.417650	−	0.908608i	$-0.637146\pi$
$241$	−12.9673	−0.835299	−0.417650	+	0.908608i	$0.637146\pi$
$242$	0	0
$243$	11.7627	0.754577
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.67572	0.106623
$248$	0	0
$249$	−21.7388	−1.37764
$250$	0	0
$251$	3.01121	0.190066	0.0950328	−	0.995474i	$-0.469704\pi$
$251$	3.01121	0.190066	0.0950328	+	0.995474i	$0.469704\pi$
$252$	0	0
$253$	−6.54241	−0.411317
$254$	0	0
$255$	−4.98210	−0.311991
$256$	0	0
$257$	−23.8489	−1.48765	−0.743827	−	0.668373i	$-0.766991\pi$
$257$	−23.8489	−1.48765	−0.743827	+	0.668373i	$0.766991\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.74415	−0.231757
$262$	0	0
$263$	21.3699	1.31773	0.658863	−	0.752263i	$-0.271038\pi$
$263$	21.3699	1.31773	0.658863	+	0.752263i	$0.271038\pi$
$264$	0	0
$265$	−0.856855	−0.0526362
$266$	0	0
$267$	−11.4637	−0.701570
$268$	0	0
$269$	−0.375658	−0.0229042	−0.0114521	−	0.999934i	$-0.503645\pi$
$269$	−0.375658	−0.0229042	−0.0114521	+	0.999934i	$0.503645\pi$
$270$	0	0
$271$	−27.1150	−1.64712	−0.823558	−	0.567232i	$-0.808014\pi$
$271$	−27.1150	−1.64712	−0.823558	+	0.567232i	$0.808014\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.76279	−0.226905
$276$	0	0
$277$	−9.85217	−0.591960	−0.295980	−	0.955194i	$-0.595646\pi$
$277$	−9.85217	−0.591960	−0.295980	+	0.955194i	$0.595646\pi$
$278$	0	0
$279$	2.53016	0.151477
$280$	0	0
$281$	−16.7895	−1.00158	−0.500788	−	0.865570i	$-0.666956\pi$
$281$	−16.7895	−1.00158	−0.500788	+	0.865570i	$0.666956\pi$
$282$	0	0
$283$	−15.3137	−0.910304	−0.455152	−	0.890414i	$-0.650415\pi$
$283$	−15.3137	−0.910304	−0.455152	+	0.890414i	$0.650415\pi$
$284$	0	0
$285$	−6.53485	−0.387091
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−5.72038	−0.336493
$290$	0	0
$291$	−6.01682	−0.352712
$292$	0	0
$293$	−9.31349	−0.544100	−0.272050	−	0.962283i	$-0.587702\pi$
$293$	−9.31349	−0.544100	−0.272050	+	0.962283i	$0.587702\pi$
$294$	0	0
$295$	−7.48416	−0.435745
$296$	0	0
$297$	−5.62984	−0.326677
$298$	0	0
$299$	−2.48867	−0.143924
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−21.5834	−1.23994
$304$	0	0
$305$	13.2894	0.760946
$306$	0	0
$307$	19.3136	1.10229	0.551143	−	0.834411i	$-0.314192\pi$
$307$	19.3136	1.10229	0.551143	+	0.834411i	$0.314192\pi$
$308$	0	0
$309$	13.7604	0.782803
$310$	0	0
$311$	6.98311	0.395976	0.197988	−	0.980204i	$-0.436559\pi$
$311$	6.98311	0.395976	0.197988	+	0.980204i	$0.436559\pi$
$312$	0	0
$313$	−19.4354	−1.09855	−0.549276	−	0.835641i	$-0.685096\pi$
$313$	−19.4354	−1.09855	−0.549276	+	0.835641i	$0.685096\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.5817	1.77381	0.886903	−	0.461956i	$-0.152852\pi$
$317$	31.5817	1.77381	0.886903	+	0.461956i	$0.152852\pi$
$318$	0	0
$319$	3.06556	0.171639
$320$	0	0
$321$	−4.50794	−0.251609
$322$	0	0
$323$	14.7951	0.823221
$324$	0	0
$325$	−1.43133	−0.0793961
$326$	0	0
$327$	9.46121	0.523206
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−18.6972	−1.02769	−0.513847	−	0.857882i	$-0.671780\pi$
$331$	−18.6972	−1.02769	−0.513847	+	0.857882i	$0.671780\pi$
$332$	0	0
$333$	−5.95819	−0.326507
$334$	0	0
$335$	8.36283	0.456910
$336$	0	0
$337$	5.71941	0.311556	0.155778	−	0.987792i	$-0.450212\pi$
$337$	5.71941	0.311556	0.155778	+	0.987792i	$0.450212\pi$
$338$	0	0
$339$	5.35502	0.290845
$340$	0	0
$341$	−2.07159	−0.112183
$342$	0	0
$343$	0	0
$344$	0	0
$345$	9.70516	0.522508
$346$	0	0
$347$	7.98329	0.428565	0.214283	−	0.976772i	$-0.431259\pi$
$347$	7.98329	0.428565	0.214283	+	0.976772i	$0.431259\pi$
$348$	0	0
$349$	−3.04781	−0.163146	−0.0815729	−	0.996667i	$-0.525994\pi$
$349$	−3.04781	−0.163146	−0.0815729	+	0.996667i	$0.525994\pi$
$350$	0	0
$351$	−2.14154	−0.114307
$352$	0	0
$353$	−31.1863	−1.65988	−0.829940	−	0.557852i	$-0.811625\pi$
$353$	−31.1863	−1.65988	−0.829940	+	0.557852i	$0.811625\pi$
$354$	0	0
$355$	−4.56678	−0.242379
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.1117	0.797564	0.398782	−	0.917046i	$-0.369433\pi$
$359$	15.1117	0.797564	0.398782	+	0.917046i	$0.369433\pi$
$360$	0	0
$361$	0.406220	0.0213800
$362$	0	0
$363$	1.33366	0.0699988
$364$	0	0
$365$	−5.97165	−0.312570
$366$	0	0
$367$	−27.8308	−1.45276	−0.726379	−	0.687295i	$-0.758798\pi$
$367$	−27.8308	−1.45276	−0.726379	+	0.687295i	$0.758798\pi$
$368$	0	0
$369$	8.01734	0.417366
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−8.85012	−0.458242	−0.229121	−	0.973398i	$-0.573585\pi$
$373$	−8.85012	−0.458242	−0.229121	+	0.973398i	$0.573585\pi$
$374$	0	0
$375$	12.9989	0.671262
$376$	0	0
$377$	1.16611	0.0600579
$378$	0	0
$379$	−26.1533	−1.34341	−0.671703	−	0.740821i	$-0.734437\pi$
$379$	−26.1533	−1.34341	−0.671703	+	0.740821i	$0.734437\pi$
$380$	0	0
$381$	−13.1549	−0.673948
$382$	0	0
$383$	−15.5873	−0.796475	−0.398237	−	0.917282i	$-0.630378\pi$
$383$	−15.5873	−0.796475	−0.398237	+	0.917282i	$0.630378\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	15.1638	0.770818
$388$	0	0
$389$	−33.3031	−1.68853	−0.844266	−	0.535924i	$-0.819963\pi$
$389$	−33.3031	−1.68853	−0.844266	+	0.535924i	$0.819963\pi$
$390$	0	0
$391$	−21.9728	−1.11121
$392$	0	0
$393$	−7.79100	−0.393004
$394$	0	0
$395$	−11.7956	−0.593502
$396$	0	0
$397$	11.3551	0.569896	0.284948	−	0.958543i	$-0.408024\pi$
$397$	11.3551	0.569896	0.284948	+	0.958543i	$0.408024\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	25.0119	1.24904	0.624518	−	0.781010i	$-0.285295\pi$
$401$	25.0119	1.24904	0.624518	+	0.781010i	$0.285295\pi$
$402$	0	0
$403$	−0.788015	−0.0392538
$404$	0	0
$405$	4.27590	0.212471
$406$	0	0
$407$	4.87833	0.241810
$408$	0	0
$409$	−2.68424	−0.132727	−0.0663636	−	0.997796i	$-0.521140\pi$
$409$	−2.68424	−0.132727	−0.0663636	+	0.997796i	$0.521140\pi$
$410$	0	0
$411$	5.33448	0.263131
$412$	0	0
$413$	0	0
$414$	0	0
$415$	18.1306	0.889997
$416$	0	0
$417$	1.76038	0.0862063
$418$	0	0
$419$	2.70453	0.132125	0.0660624	−	0.997815i	$-0.478956\pi$
$419$	2.70453	0.132125	0.0660624	+	0.997815i	$0.478956\pi$
$420$	0	0
$421$	−14.3434	−0.699057	−0.349528	−	0.936926i	$-0.613658\pi$
$421$	−14.3434	−0.699057	−0.349528	+	0.936926i	$0.613658\pi$
$422$	0	0
$423$	−8.98739	−0.436982
$424$	0	0
$425$	−12.6374	−0.613004
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.507311	0.0244932
$430$	0	0
$431$	34.0273	1.63904	0.819520	−	0.573051i	$-0.194240\pi$
$431$	34.0273	1.63904	0.819520	+	0.573051i	$0.194240\pi$
$432$	0	0
$433$	−29.4843	−1.41692	−0.708462	−	0.705749i	$-0.750611\pi$
$433$	−29.4843	−1.41692	−0.708462	+	0.705749i	$0.750611\pi$
$434$	0	0
$435$	−4.54753	−0.218037
$436$	0	0
$437$	−28.8209	−1.37869
$438$	0	0
$439$	−24.9457	−1.19059	−0.595297	−	0.803506i	$-0.702966\pi$
$439$	−24.9457	−1.19059	−0.595297	+	0.803506i	$0.702966\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	37.8336	1.79753	0.898765	−	0.438431i	$-0.144466\pi$
$443$	37.8336	1.79753	0.898765	+	0.438431i	$0.144466\pi$
$444$	0	0
$445$	9.56100	0.453235
$446$	0	0
$447$	−5.64479	−0.266989
$448$	0	0
$449$	27.5777	1.30147	0.650736	−	0.759304i	$-0.274461\pi$
$449$	27.5777	1.30147	0.650736	+	0.759304i	$0.274461\pi$
$450$	0	0
$451$	−6.56428	−0.309100
$452$	0	0
$453$	30.9795	1.45554
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.2292	1.50762	0.753810	−	0.657093i	$-0.228214\pi$
$457$	32.2292	1.50762	0.753810	+	0.657093i	$0.228214\pi$
$458$	0	0
$459$	−18.9079	−0.882546
$460$	0	0
$461$	−20.1755	−0.939668	−0.469834	−	0.882755i	$-0.655686\pi$
$461$	−20.1755	−0.939668	−0.469834	+	0.882755i	$0.655686\pi$
$462$	0	0
$463$	5.41828	0.251809	0.125904	−	0.992042i	$-0.459817\pi$
$463$	5.41828	0.251809	0.125904	+	0.992042i	$0.459817\pi$
$464$	0	0
$465$	3.07305	0.142509
$466$	0	0
$467$	33.6840	1.55871	0.779356	−	0.626582i	$-0.215547\pi$
$467$	33.6840	1.55871	0.779356	+	0.626582i	$0.215547\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−4.75223	−0.218971
$472$	0	0
$473$	−12.4155	−0.570865
$474$	0	0
$475$	−16.5760	−0.760561
$476$	0	0
$477$	−0.940870	−0.0430795
$478$	0	0
$479$	25.1407	1.14871	0.574354	−	0.818607i	$-0.305253\pi$
$479$	25.1407	1.14871	0.574354	+	0.818607i	$0.305253\pi$
$480$	0	0
$481$	1.85567	0.0846114
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.01815	0.227863
$486$	0	0
$487$	−29.2345	−1.32474	−0.662372	−	0.749175i	$-0.730450\pi$
$487$	−29.2345	−1.32474	−0.662372	+	0.749175i	$0.730450\pi$
$488$	0	0
$489$	29.3108	1.32548
$490$	0	0
$491$	17.8960	0.807634	0.403817	−	0.914840i	$-0.367683\pi$
$491$	17.8960	0.807634	0.403817	+	0.914840i	$0.367683\pi$
$492$	0	0
$493$	10.2957	0.463697
$494$	0	0
$495$	1.35852	0.0610607
$496$	0	0
$497$	0	0
$498$	0	0
$499$	21.2061	0.949316	0.474658	−	0.880170i	$-0.342572\pi$
$499$	21.2061	0.949316	0.474658	+	0.880170i	$0.342572\pi$
$500$	0	0
$501$	8.34267	0.372723
$502$	0	0
$503$	14.2236	0.634200	0.317100	−	0.948392i	$-0.397291\pi$
$503$	14.2236	0.634200	0.317100	+	0.948392i	$0.397291\pi$
$504$	0	0
$505$	18.0010	0.801035
$506$	0	0
$507$	−17.1446	−0.761417
$508$	0	0
$509$	−8.35639	−0.370391	−0.185195	−	0.982702i	$-0.559292\pi$
$509$	−8.35639	−0.370391	−0.185195	+	0.982702i	$0.559292\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−24.8009	−1.09499
$514$	0	0
$515$	−11.4765	−0.505714
$516$	0	0
$517$	7.35852	0.323627
$518$	0	0
$519$	−0.582540	−0.0255707
$520$	0	0
$521$	−15.4460	−0.676700	−0.338350	−	0.941020i	$-0.609869\pi$
$521$	−15.4460	−0.676700	−0.338350	+	0.941020i	$0.609869\pi$
$522$	0	0
$523$	12.2845	0.537163	0.268581	−	0.963257i	$-0.413445\pi$
$523$	12.2845	0.537163	0.268581	+	0.963257i	$0.413445\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.95747	−0.303072
$528$	0	0
$529$	19.8031	0.861003
$530$	0	0
$531$	−8.21799	−0.356630
$532$	0	0
$533$	−2.49699	−0.108157
$534$	0	0
$535$	3.75972	0.162547
$536$	0	0
$537$	−10.9172	−0.471111
$538$	0	0
$539$	0	0
$540$	0	0
$541$	11.4726	0.493244	0.246622	−	0.969112i	$-0.420679\pi$
$541$	11.4726	0.493244	0.246622	+	0.969112i	$0.420679\pi$
$542$	0	0
$543$	−24.7509	−1.06216
$544$	0	0
$545$	−7.89085	−0.338007
$546$	0	0
$547$	4.76618	0.203787	0.101893	−	0.994795i	$-0.467510\pi$
$547$	4.76618	0.203787	0.101893	+	0.994795i	$0.467510\pi$
$548$	0	0
$549$	14.5924	0.622787
$550$	0	0
$551$	13.5046	0.575314
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−7.23663	−0.307178
$556$	0	0
$557$	15.7820	0.668704	0.334352	−	0.942448i	$-0.391483\pi$
$557$	15.7820	0.668704	0.334352	+	0.942448i	$0.391483\pi$
$558$	0	0
$559$	−4.72274	−0.199751
$560$	0	0
$561$	4.47911	0.189108
$562$	0	0
$563$	11.3866	0.479888	0.239944	−	0.970787i	$-0.422871\pi$
$563$	11.3866	0.479888	0.239944	+	0.970787i	$0.422871\pi$
$564$	0	0
$565$	−4.46620	−0.187895
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.6468	−0.781714	−0.390857	−	0.920451i	$-0.627821\pi$
$569$	−18.6468	−0.781714	−0.390857	+	0.920451i	$0.627821\pi$
$570$	0	0
$571$	−17.4595	−0.730656	−0.365328	−	0.930879i	$-0.619043\pi$
$571$	−17.4595	−0.730656	−0.365328	+	0.930879i	$0.619043\pi$
$572$	0	0
$573$	−1.60238	−0.0669405
$574$	0	0
$575$	24.6177	1.02663
$576$	0	0
$577$	18.1981	0.757596	0.378798	−	0.925479i	$-0.376338\pi$
$577$	18.1981	0.757596	0.378798	+	0.925479i	$0.376338\pi$
$578$	0	0
$579$	−27.2950	−1.13434
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0.770347	0.0319045
$584$	0	0
$585$	0.516767	0.0213657
$586$	0	0
$587$	−35.2867	−1.45644	−0.728218	−	0.685346i	$-0.759651\pi$
$587$	−35.2867	−1.45644	−0.728218	+	0.685346i	$0.759651\pi$
$588$	0	0
$589$	−9.12588	−0.376025
$590$	0	0
$591$	−20.6384	−0.848949
$592$	0	0
$593$	38.5834	1.58443	0.792214	−	0.610243i	$-0.208928\pi$
$593$	38.5834	1.58443	0.792214	+	0.610243i	$0.208928\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−17.5703	−0.719104
$598$	0	0
$599$	−40.2802	−1.64580	−0.822902	−	0.568183i	$-0.807647\pi$
$599$	−40.2802	−1.64580	−0.822902	+	0.568183i	$0.807647\pi$
$600$	0	0
$601$	−2.55263	−0.104124	−0.0520620	−	0.998644i	$-0.516579\pi$
$601$	−2.55263	−0.104124	−0.0520620	+	0.998644i	$0.516579\pi$
$602$	0	0
$603$	9.18281	0.373953
$604$	0	0
$605$	−1.11230	−0.0452213
$606$	0	0
$607$	−11.5613	−0.469259	−0.234629	−	0.972085i	$-0.575388\pi$
$607$	−11.5613	−0.469259	−0.234629	+	0.972085i	$0.575388\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.79911	0.113240
$612$	0	0
$613$	35.6688	1.44065	0.720324	−	0.693638i	$-0.243993\pi$
$613$	35.6688	1.44065	0.720324	+	0.693638i	$0.243993\pi$
$614$	0	0
$615$	9.73761	0.392658
$616$	0	0
$617$	−21.9721	−0.884565	−0.442283	−	0.896876i	$-0.645831\pi$
$617$	−21.9721	−0.884565	−0.442283	+	0.896876i	$0.645831\pi$
$618$	0	0
$619$	9.61296	0.386378	0.193189	−	0.981162i	$-0.438117\pi$
$619$	9.61296	0.386378	0.193189	+	0.981162i	$0.438117\pi$
$620$	0	0
$621$	36.8327	1.47805
$622$	0	0
$623$	0	0
$624$	0	0
$625$	7.97258	0.318903
$626$	0	0
$627$	5.87509	0.234629
$628$	0	0
$629$	16.3839	0.653270
$630$	0	0
$631$	19.8407	0.789846	0.394923	−	0.918714i	$-0.370771\pi$
$631$	19.8407	0.789846	0.394923	+	0.918714i	$0.370771\pi$
$632$	0	0
$633$	−17.6048	−0.699729
$634$	0	0
$635$	10.9715	0.435391
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−5.01455	−0.198373
$640$	0	0
$641$	17.3313	0.684546	0.342273	−	0.939601i	$-0.388803\pi$
$641$	17.3313	0.684546	0.342273	+	0.939601i	$0.388803\pi$
$642$	0	0
$643$	24.6406	0.971731	0.485865	−	0.874034i	$-0.338505\pi$
$643$	24.6406	0.971731	0.485865	+	0.874034i	$0.338505\pi$
$644$	0	0
$645$	18.4174	0.725185
$646$	0	0
$647$	−19.8863	−0.781813	−0.390906	−	0.920430i	$-0.627838\pi$
$647$	−19.8863	−0.781813	−0.390906	+	0.920430i	$0.627838\pi$
$648$	0	0
$649$	6.72856	0.264119
$650$	0	0
$651$	0	0
$652$	0	0
$653$	31.4742	1.23168	0.615841	−	0.787871i	$-0.288817\pi$
$653$	31.4742	1.23168	0.615841	+	0.787871i	$0.288817\pi$
$654$	0	0
$655$	6.49786	0.253892
$656$	0	0
$657$	−6.55717	−0.255820
$658$	0	0
$659$	1.34601	0.0524332	0.0262166	−	0.999656i	$-0.491654\pi$
$659$	1.34601	0.0524332	0.0262166	+	0.999656i	$0.491654\pi$
$660$	0	0
$661$	2.30467	0.0896412	0.0448206	−	0.998995i	$-0.485728\pi$
$661$	2.30467	0.0896412	0.0448206	+	0.998995i	$0.485728\pi$
$662$	0	0
$663$	1.70381	0.0661706
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.0562	−0.776578
$668$	0	0
$669$	−12.6663	−0.489708
$670$	0	0
$671$	−11.9477	−0.461234
$672$	0	0
$673$	13.3286	0.513779	0.256889	−	0.966441i	$-0.417302\pi$
$673$	13.3286	0.513779	0.256889	+	0.966441i	$0.417302\pi$
$674$	0	0
$675$	21.1839	0.815370
$676$	0	0
$677$	48.7092	1.87205	0.936024	−	0.351937i	$-0.114477\pi$
$677$	48.7092	1.87205	0.936024	+	0.351937i	$0.114477\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	4.30750	0.165064
$682$	0	0
$683$	−8.48938	−0.324837	−0.162419	−	0.986722i	$-0.551929\pi$
$683$	−8.48938	−0.324837	−0.162419	+	0.986722i	$0.551929\pi$
$684$	0	0
$685$	−4.44907	−0.169990
$686$	0	0
$687$	14.4095	0.549755
$688$	0	0
$689$	0.293033	0.0111637
$690$	0	0
$691$	−17.2754	−0.657187	−0.328593	−	0.944471i	$-0.606575\pi$
$691$	−17.2754	−0.657187	−0.328593	+	0.944471i	$0.606575\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.46820	−0.0556918
$696$	0	0
$697$	−22.0462	−0.835061
$698$	0	0
$699$	−35.9346	−1.35917
$700$	0	0
$701$	−15.2340	−0.575382	−0.287691	−	0.957723i	$-0.592888\pi$
$701$	−15.2340	−0.575382	−0.287691	+	0.957723i	$0.592888\pi$
$702$	0	0
$703$	21.4903	0.810520
$704$	0	0
$705$	−10.9158	−0.411113
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8.32377	0.312606	0.156303	−	0.987709i	$-0.450042\pi$
$709$	8.32377	0.312606	0.156303	+	0.987709i	$0.450042\pi$
$710$	0	0
$711$	−12.9522	−0.485745
$712$	0	0
$713$	13.5532	0.507571
$714$	0	0
$715$	−0.423108	−0.0158233
$716$	0	0
$717$	−6.42944	−0.240112
$718$	0	0
$719$	−20.8116	−0.776142	−0.388071	−	0.921629i	$-0.626859\pi$
$719$	−20.8116	−0.776142	−0.388071	+	0.921629i	$0.626859\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−17.2940	−0.643170
$724$	0	0
$725$	−11.5351	−0.428402
$726$	0	0
$727$	−28.0523	−1.04040	−0.520202	−	0.854043i	$-0.674143\pi$
$727$	−28.0523	−1.04040	−0.520202	+	0.854043i	$0.674143\pi$
$728$	0	0
$729$	27.2200	1.00815
$730$	0	0
$731$	−41.6976	−1.54224
$732$	0	0
$733$	−28.8149	−1.06430	−0.532152	−	0.846649i	$-0.678617\pi$
$733$	−28.8149	−1.06430	−0.532152	+	0.846649i	$0.678617\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.51852	−0.276948
$738$	0	0
$739$	26.0621	0.958709	0.479355	−	0.877621i	$-0.340871\pi$
$739$	26.0621	0.958709	0.479355	+	0.877621i	$0.340871\pi$
$740$	0	0
$741$	2.23483	0.0820986
$742$	0	0
$743$	−11.2002	−0.410894	−0.205447	−	0.978668i	$-0.565865\pi$
$743$	−11.2002	−0.410894	−0.205447	+	0.978668i	$0.565865\pi$
$744$	0	0
$745$	4.70787	0.172483
$746$	0	0
$747$	19.9083	0.728408
$748$	0	0
$749$	0	0
$750$	0	0
$751$	35.1144	1.28134	0.640672	−	0.767815i	$-0.278656\pi$
$751$	35.1144	1.28134	0.640672	+	0.767815i	$0.278656\pi$
$752$	0	0
$753$	4.01592	0.146348
$754$	0	0
$755$	−25.8375	−0.940324
$756$	0	0
$757$	3.93638	0.143070	0.0715350	−	0.997438i	$-0.477210\pi$
$757$	3.93638	0.143070	0.0715350	+	0.997438i	$0.477210\pi$
$758$	0	0
$759$	−8.72532	−0.316709
$760$	0	0
$761$	−25.7784	−0.934466	−0.467233	−	0.884134i	$-0.654749\pi$
$761$	−25.7784	−0.934466	−0.467233	+	0.884134i	$0.654749\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	4.56259	0.164961
$766$	0	0
$767$	2.55948	0.0924176
$768$	0	0
$769$	−14.3551	−0.517657	−0.258828	−	0.965923i	$-0.583336\pi$
$769$	−14.3551	−0.517657	−0.258828	+	0.965923i	$0.583336\pi$
$770$	0	0
$771$	−31.8063	−1.14547
$772$	0	0
$773$	0.891084	0.0320501	0.0160250	−	0.999872i	$-0.494899\pi$
$773$	0.891084	0.0320501	0.0160250	+	0.999872i	$0.494899\pi$
$774$	0	0
$775$	7.79497	0.280004
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−28.9173	−1.03607
$780$	0	0
$781$	4.10572	0.146914
$782$	0	0
$783$	−17.2586	−0.616774
$784$	0	0
$785$	3.96346	0.141462
$786$	0	0
$787$	−47.9871	−1.71056	−0.855278	−	0.518169i	$-0.826614\pi$
$787$	−47.9871	−1.71056	−0.855278	+	0.518169i	$0.826614\pi$
$788$	0	0
$789$	28.5001	1.01463
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.54478	−0.161390
$794$	0	0
$795$	−1.14275	−0.0405292
$796$	0	0
$797$	−42.2980	−1.49827	−0.749136	−	0.662416i	$-0.769531\pi$
$797$	−42.2980	−1.49827	−0.749136	+	0.662416i	$0.769531\pi$
$798$	0	0
$799$	24.7137	0.874307
$800$	0	0
$801$	10.4985	0.370945
$802$	0	0
$803$	5.36875	0.189459
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.500998	−0.0176360
$808$	0	0
$809$	−50.9993	−1.79304	−0.896520	−	0.443003i	$-0.853913\pi$
$809$	−50.9993	−1.79304	−0.896520	+	0.443003i	$0.853913\pi$
$810$	0	0
$811$	20.7203	0.727589	0.363794	−	0.931479i	$-0.381481\pi$
$811$	20.7203	0.727589	0.363794	+	0.931479i	$0.381481\pi$
$812$	0	0
$813$	−36.1621	−1.26826
$814$	0	0
$815$	−24.4458	−0.856300
$816$	0	0
$817$	−54.6933	−1.91348
$818$	0	0
$819$	0	0
$820$	0	0
$821$	23.7097	0.827473	0.413736	−	0.910397i	$-0.364224\pi$
$821$	23.7097	0.827473	0.413736	+	0.910397i	$0.364224\pi$
$822$	0	0
$823$	6.71635	0.234117	0.117059	−	0.993125i	$-0.462653\pi$
$823$	6.71635	0.234117	0.117059	+	0.993125i	$0.462653\pi$
$824$	0	0
$825$	−5.01828	−0.174714
$826$	0	0
$827$	32.4414	1.12810	0.564048	−	0.825742i	$-0.309243\pi$
$827$	32.4414	1.12810	0.564048	+	0.825742i	$0.309243\pi$
$828$	0	0
$829$	35.5039	1.23310	0.616551	−	0.787315i	$-0.288530\pi$
$829$	35.5039	1.23310	0.616551	+	0.787315i	$0.288530\pi$
$830$	0	0
$831$	−13.1394	−0.455801
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−6.95796	−0.240790
$836$	0	0
$837$	11.6627	0.403123
$838$	0	0
$839$	46.9126	1.61960	0.809802	−	0.586704i	$-0.199575\pi$
$839$	46.9126	1.61960	0.809802	+	0.586704i	$0.199575\pi$
$840$	0	0
$841$	−19.6023	−0.675942
$842$	0	0
$843$	−22.3914	−0.771201
$844$	0	0
$845$	14.2989	0.491898
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−20.4232	−0.700922
$850$	0	0
$851$	−31.9160	−1.09407
$852$	0	0
$853$	36.1884	1.23907	0.619533	−	0.784971i	$-0.287322\pi$
$853$	36.1884	1.23907	0.619533	+	0.784971i	$0.287322\pi$
$854$	0	0
$855$	5.98460	0.204669
$856$	0	0
$857$	23.3096	0.796240	0.398120	−	0.917333i	$-0.369663\pi$
$857$	23.3096	0.796240	0.398120	+	0.917333i	$0.369663\pi$
$858$	0	0
$859$	−51.3637	−1.75251	−0.876254	−	0.481849i	$-0.839965\pi$
$859$	−51.3637	−1.75251	−0.876254	+	0.481849i	$0.839965\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26.8102	0.912629	0.456314	−	0.889819i	$-0.349169\pi$
$863$	26.8102	0.912629	0.456314	+	0.889819i	$0.349169\pi$
$864$	0	0
$865$	0.485850	0.0165194
$866$	0	0
$867$	−7.62902	−0.259095
$868$	0	0
$869$	10.6047	0.359741
$870$	0	0
$871$	−2.85998	−0.0969066
$872$	0	0
$873$	5.51018	0.186491
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14.3717	0.485298	0.242649	−	0.970114i	$-0.421984\pi$
$877$	14.3717	0.485298	0.242649	+	0.970114i	$0.421984\pi$
$878$	0	0
$879$	−12.4210	−0.418950
$880$	0	0
$881$	−41.8248	−1.40911	−0.704557	−	0.709648i	$-0.748854\pi$
$881$	−41.8248	−1.40911	−0.704557	+	0.709648i	$0.748854\pi$
$882$	0	0
$883$	27.4995	0.925431	0.462715	−	0.886507i	$-0.346875\pi$
$883$	27.4995	0.925431	0.462715	+	0.886507i	$0.346875\pi$
$884$	0	0
$885$	−9.98131	−0.335518
$886$	0	0
$887$	−27.3796	−0.919316	−0.459658	−	0.888096i	$-0.652028\pi$
$887$	−27.3796	−0.919316	−0.459658	+	0.888096i	$0.652028\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3.84420	−0.128786
$892$	0	0
$893$	32.4161	1.08476
$894$	0	0
$895$	9.10516	0.304352
$896$	0	0
$897$	−3.31903	−0.110819
$898$	0	0
$899$	−6.35060	−0.211804
$900$	0	0
$901$	2.58722	0.0861928
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.6428	0.686189
$906$	0	0
$907$	−2.30348	−0.0764858	−0.0382429	−	0.999268i	$-0.512176\pi$
$907$	−2.30348	−0.0764858	−0.0382429	+	0.999268i	$0.512176\pi$
$908$	0	0
$909$	19.7660	0.655598
$910$	0	0
$911$	16.1674	0.535651	0.267826	−	0.963467i	$-0.413695\pi$
$911$	16.1674	0.535651	0.267826	+	0.963467i	$0.413695\pi$
$912$	0	0
$913$	−16.3001	−0.539456
$914$	0	0
$915$	17.7234	0.585919
$916$	0	0
$917$	0	0
$918$	0	0
$919$	40.6181	1.33987	0.669935	−	0.742420i	$-0.266322\pi$
$919$	40.6181	1.33987	0.669935	+	0.742420i	$0.266322\pi$
$920$	0	0
$921$	25.7577	0.848746
$922$	0	0
$923$	1.56178	0.0514065
$924$	0	0
$925$	−18.3561	−0.603546
$926$	0	0
$927$	−12.6018	−0.413896
$928$	0	0
$929$	−20.6241	−0.676656	−0.338328	−	0.941028i	$-0.609861\pi$
$929$	−20.6241	−0.676656	−0.338328	+	0.941028i	$0.609861\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	9.31308	0.304896
$934$	0	0
$935$	−3.73567	−0.122169
$936$	0	0
$937$	43.6953	1.42746	0.713732	−	0.700419i	$-0.247003\pi$
$937$	43.6953	1.42746	0.713732	+	0.700419i	$0.247003\pi$
$938$	0	0
$939$	−25.9201	−0.845870
$940$	0	0
$941$	60.3713	1.96805	0.984025	−	0.178032i	$-0.0569731\pi$
$941$	60.3713	1.96805	0.984025	+	0.178032i	$0.0569731\pi$
$942$	0	0
$943$	42.9462	1.39852
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.8924	−1.29633	−0.648165	−	0.761500i	$-0.724463\pi$
$947$	−39.8924	−1.29633	−0.648165	+	0.761500i	$0.724463\pi$
$948$	0	0
$949$	2.04222	0.0662934
$950$	0	0
$951$	42.1192	1.36581
$952$	0	0
$953$	27.3714	0.886648	0.443324	−	0.896362i	$-0.353799\pi$
$953$	27.3714	0.886648	0.443324	+	0.896362i	$0.353799\pi$
$954$	0	0
$955$	1.33642	0.0432455
$956$	0	0
$957$	4.08841	0.132160
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−26.7085	−0.861565
$962$	0	0
$963$	4.12836	0.133035
$964$	0	0
$965$	22.7646	0.732818
$966$	0	0
$967$	28.6973	0.922844	0.461422	−	0.887181i	$-0.347339\pi$
$967$	28.6973	0.922844	0.461422	+	0.887181i	$0.347339\pi$
$968$	0	0
$969$	19.7316	0.633870
$970$	0	0
$971$	15.7663	0.505966	0.252983	−	0.967471i	$-0.418588\pi$
$971$	15.7663	0.505966	0.252983	+	0.967471i	$0.418588\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1.90891	−0.0611339
$976$	0	0
$977$	−29.7736	−0.952543	−0.476271	−	0.879298i	$-0.658012\pi$
$977$	−29.7736	−0.952543	−0.476271	+	0.879298i	$0.658012\pi$
$978$	0	0
$979$	−8.59572	−0.274720
$980$	0	0
$981$	−8.66455	−0.276638
$982$	0	0
$983$	48.8969	1.55957	0.779785	−	0.626048i	$-0.215329\pi$
$983$	48.8969	1.55957	0.779785	+	0.626048i	$0.215329\pi$
$984$	0	0
$985$	17.2128	0.548446
$986$	0	0
$987$	0	0
$988$	0	0
$989$	81.2271	2.58287
$990$	0	0
$991$	−1.71293	−0.0544131	−0.0272065	−	0.999630i	$-0.508661\pi$
$991$	−1.71293	−0.0544131	−0.0272065	+	0.999630i	$0.508661\pi$
$992$	0	0
$993$	−24.9357	−0.791311
$994$	0	0
$995$	14.6540	0.464563
$996$	0	0
$997$	21.6116	0.684447	0.342223	−	0.939619i	$-0.388820\pi$
$997$	21.6116	0.684447	0.342223	+	0.939619i	$0.388820\pi$
$998$	0	0
$999$	−27.4642	−0.868930

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.db.1.4		5
4.3	odd	2		4312.2.a.bg.1.2		5
7.3	odd	6		1232.2.q.o.177.4		10
7.5	odd	6		1232.2.q.o.529.4		10
7.6	odd	2		8624.2.a.dc.1.2		5
28.3	even	6		616.2.q.f.177.2	✓	10
28.19	even	6		616.2.q.f.529.2	yes	10
28.27	even	2		4312.2.a.bf.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.q.f.177.2	✓	10	28.3	even	6
616.2.q.f.529.2	yes	10	28.19	even	6
1232.2.q.o.177.4		10	7.3	odd	6
1232.2.q.o.529.4		10	7.5	odd	6
4312.2.a.bf.1.4		5	28.27	even	2
4312.2.a.bg.1.2		5	4.3	odd	2
8624.2.a.db.1.4		5	1.1	even	1	trivial
8624.2.a.dc.1.2		5	7.6	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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q+1.33366 q^{3} -1.11230 q^{5} -1.22136 q^{9} +1.00000 q^{11} +0.380391 q^{13} -1.48342 q^{15} +3.35852 q^{17} +4.40525 q^{19} -6.54241 q^{23} -3.76279 q^{25} -5.62984 q^{27} +3.06556 q^{29} -2.07159 q^{31} +1.33366 q^{33} +4.87833 q^{37} +0.507311 q^{39} -6.56428 q^{41} -12.4155 q^{43} +1.35852 q^{45} +7.35852 q^{47} +4.47911 q^{51} +0.770347 q^{53} -1.11230 q^{55} +5.87509 q^{57} +6.72856 q^{59} -11.9477 q^{61} -0.423108 q^{65} -7.51852 q^{67} -8.72532 q^{69} +4.10572 q^{71} +5.36875 q^{73} -5.01828 q^{75} +10.6047 q^{79} -3.84420 q^{81} -16.3001 q^{83} -3.73567 q^{85} +4.08841 q^{87} -8.59572 q^{89} -2.76279 q^{93} -4.89995 q^{95} -4.51152 q^{97} -1.22136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - 4 q^{5} + 5 q^{11} - q^{13} - 7 q^{15} - 9 q^{17} + q^{19} + 8 q^{23} + 5 q^{25} - 4 q^{27} + 9 q^{29} + 3 q^{31} - q^{33} + 2 q^{37} + 7 q^{39} - 15 q^{41} - 14 q^{43} - 19 q^{45} + 11 q^{47}+ \cdots - 7 q^{97}+O(q^{100}) \) 5 * q - q^3 - 4 * q^5 + 5 * q^11 - q^13 - 7 * q^15 - 9 * q^17 + q^19 + 8 * q^23 + 5 * q^25 - 4 * q^27 + 9 * q^29 + 3 * q^31 - q^33 + 2 * q^37 + 7 * q^39 - 15 * q^41 - 14 * q^43 - 19 * q^45 + 11 * q^47 + 14 * q^51 - 9 * q^53 - 4 * q^55 + 4 * q^57 + 4 * q^59 + 2 * q^61 + 7 * q^65 + 8 * q^67 - 11 * q^69 - 15 * q^71 - 26 * q^73 + 27 * q^75 + 3 * q^79 - 19 * q^81 - q^83 + 13 * q^85 + 14 * q^87 - 41 * q^89 + 10 * q^93 + 19 * q^95 - 7 * q^97

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