LMFDB - Embedded newform 8624.2.a.cn.1.3

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 = [3,0,1,0,2,0,0,0,4,0,3,0,-3,0,15,0,5,0,11,0,0,0,4,0,11,0,22,0, -1,0,19,0,1,0,0,0,-8] 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:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 308)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.239123$ 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)

$f(q)$	$=$	$q+3.18194 q^{3} +3.42107 q^{5} +7.12476 q^{9} +1.00000 q^{11} -3.66019 q^{13} +10.8856 q^{15} +1.76088 q^{17} +3.76088 q^{19} +1.23912 q^{23} +6.70370 q^{25} +13.1248 q^{27} -2.89931 q^{29} +9.46457 q^{31} +3.18194 q^{33} +2.84213 q^{37} -11.6465 q^{39} -8.70370 q^{41} -12.0676 q^{43} +24.3743 q^{45} +4.23912 q^{47} +5.60301 q^{51} -0.225450 q^{53} +3.42107 q^{55} +11.9669 q^{57} -6.54583 q^{59} +10.2495 q^{61} -12.5218 q^{65} +3.40739 q^{67} +3.94282 q^{69} +4.23912 q^{71} -10.1819 q^{73} +21.3308 q^{75} +7.35021 q^{79} +20.3880 q^{81} -10.4887 q^{83} +6.02408 q^{85} -9.22545 q^{87} -7.26320 q^{89} +30.1157 q^{93} +12.8662 q^{95} -11.7472 q^{97} +7.12476 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 2 q^{5} + 4 q^{9} + 3 q^{11} - 3 q^{13} + 15 q^{15} + 5 q^{17} + 11 q^{19} + 4 q^{23} + 11 q^{25} + 22 q^{27} - q^{29} + 19 q^{31} + q^{33} - 8 q^{37} - 17 q^{39} - 17 q^{41} - 10 q^{43}+ \cdots + 4 q^{99}+O(q^{100}) $ 3 * q + q^3 + 2 * q^5 + 4 * q^9 + 3 * q^11 - 3 * q^13 + 15 * q^15 + 5 * q^17 + 11 * q^19 + 4 * q^23 + 11 * q^25 + 22 * q^27 - q^29 + 19 * q^31 + q^33 - 8 * q^37 - 17 * q^39 - 17 * q^41 - 10 * q^43 + 21 * q^45 + 13 * q^47 + 9 * q^53 + 2 * q^55 + 2 * q^57 + 6 * q^59 - 4 * q^61 - 37 * q^65 - 8 * q^67 + 3 * q^69 + 13 * q^71 - 22 * q^73 + 13 * q^75 - 5 * q^79 + 19 * q^81 + 3 * q^83 - 7 * q^85 - 18 * q^87 + 3 * q^89 + 14 * q^93 - 3 * q^95 - 25 * q^97 + 4 * q^99

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
$3$	3.18194	1.83710	0.918548	−	0.395310i	$-0.129363\pi$
$3$	3.18194	1.83710	0.918548	+	0.395310i	$0.129363\pi$
$4$	0	0
$5$	3.42107	1.52995	0.764974	−	0.644061i	$-0.222752\pi$
$5$	3.42107	1.52995	0.764974	+	0.644061i	$0.222752\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.12476	2.37492
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.66019	−1.01515	−0.507577	−	0.861606i	$-0.669459\pi$
$13$	−3.66019	−1.01515	−0.507577	+	0.861606i	$0.669459\pi$
$14$	0	0
$15$	10.8856	2.81066
$16$	0	0
$17$	1.76088	0.427075	0.213538	−	0.976935i	$-0.431501\pi$
$17$	1.76088	0.427075	0.213538	+	0.976935i	$0.431501\pi$
$18$	0	0
$19$	3.76088	0.862804	0.431402	−	0.902160i	$-0.358019\pi$
$19$	3.76088	0.862804	0.431402	+	0.902160i	$0.358019\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.23912	0.258375	0.129188	−	0.991620i	$-0.458763\pi$
$23$	1.23912	0.258375	0.129188	+	0.991620i	$0.458763\pi$
$24$	0	0
$25$	6.70370	1.34074
$26$	0	0
$27$	13.1248	2.52586
$28$	0	0
$29$	−2.89931	−0.538389	−0.269194	−	0.963086i	$-0.586757\pi$
$29$	−2.89931	−0.538389	−0.269194	+	0.963086i	$0.586757\pi$
$30$	0	0
$31$	9.46457	1.69989	0.849944	−	0.526873i	$-0.176636\pi$
$31$	9.46457	1.69989	0.849944	+	0.526873i	$0.176636\pi$
$32$	0	0
$33$	3.18194	0.553905
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.84213	0.467244	0.233622	−	0.972328i	$-0.424942\pi$
$37$	2.84213	0.467244	0.233622	+	0.972328i	$0.424942\pi$
$38$	0	0
$39$	−11.6465	−1.86494
$40$	0	0
$41$	−8.70370	−1.35929	−0.679645	−	0.733542i	$-0.737866\pi$
$41$	−8.70370	−1.35929	−0.679645	+	0.733542i	$0.737866\pi$
$42$	0	0
$43$	−12.0676	−1.84029	−0.920145	−	0.391579i	$-0.871929\pi$
$43$	−12.0676	−1.84029	−0.920145	+	0.391579i	$0.871929\pi$
$44$	0	0
$45$	24.3743	3.63350
$46$	0	0
$47$	4.23912	0.618340	0.309170	−	0.951007i	$-0.399949\pi$
$47$	4.23912	0.618340	0.309170	+	0.951007i	$0.399949\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	5.60301	0.784578
$52$	0	0
$53$	−0.225450	−0.0309680	−0.0154840	−	0.999880i	$-0.504929\pi$
$53$	−0.225450	−0.0309680	−0.0154840	+	0.999880i	$0.504929\pi$
$54$	0	0
$55$	3.42107	0.461297
$56$	0	0
$57$	11.9669	1.58505
$58$	0	0
$59$	−6.54583	−0.852194	−0.426097	−	0.904677i	$-0.640112\pi$
$59$	−6.54583	−0.852194	−0.426097	+	0.904677i	$0.640112\pi$
$60$	0	0
$61$	10.2495	1.31232	0.656159	−	0.754623i	$-0.272180\pi$
$61$	10.2495	1.31232	0.656159	+	0.754623i	$0.272180\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.5218	−1.55313
$66$	0	0
$67$	3.40739	0.416279	0.208140	−	0.978099i	$-0.433259\pi$
$67$	3.40739	0.416279	0.208140	+	0.978099i	$0.433259\pi$
$68$	0	0
$69$	3.94282	0.474660
$70$	0	0
$71$	4.23912	0.503091	0.251546	−	0.967845i	$-0.419061\pi$
$71$	4.23912	0.503091	0.251546	+	0.967845i	$0.419061\pi$
$72$	0	0
$73$	−10.1819	−1.19171	−0.595853	−	0.803093i	$-0.703186\pi$
$73$	−10.1819	−1.19171	−0.595853	+	0.803093i	$0.703186\pi$
$74$	0	0
$75$	21.3308	2.46307
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.35021	0.826964	0.413482	−	0.910512i	$-0.364313\pi$
$79$	7.35021	0.826964	0.413482	+	0.910512i	$0.364313\pi$
$80$	0	0
$81$	20.3880	2.26533
$82$	0	0
$83$	−10.4887	−1.15128	−0.575639	−	0.817704i	$-0.695247\pi$
$83$	−10.4887	−1.15128	−0.575639	+	0.817704i	$0.695247\pi$
$84$	0	0
$85$	6.02408	0.653403
$86$	0	0
$87$	−9.22545	−0.989072
$88$	0	0
$89$	−7.26320	−0.769898	−0.384949	−	0.922938i	$-0.625781\pi$
$89$	−7.26320	−0.769898	−0.384949	+	0.922938i	$0.625781\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	30.1157	3.12286
$94$	0	0
$95$	12.8662	1.32005
$96$	0	0
$97$	−11.7472	−1.19275	−0.596374	−	0.802707i	$-0.703392\pi$
$97$	−11.7472	−1.19275	−0.596374	+	0.802707i	$0.703392\pi$
$98$	0	0
$99$	7.12476	0.716066
$100$	0	0
$101$	−12.0539	−1.19941	−0.599704	−	0.800222i	$-0.704715\pi$
$101$	−12.0539	−1.19941	−0.599704	+	0.800222i	$0.704715\pi$
$102$	0	0
$103$	−16.8960	−1.66482	−0.832408	−	0.554163i	$-0.813039\pi$
$103$	−16.8960	−1.66482	−0.832408	+	0.554163i	$0.813039\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.14419	−0.110613	−0.0553067	−	0.998469i	$-0.517614\pi$
$107$	−1.14419	−0.110613	−0.0553067	+	0.998469i	$0.517614\pi$
$108$	0	0
$109$	−3.30671	−0.316725	−0.158363	−	0.987381i	$-0.550621\pi$
$109$	−3.30671	−0.316725	−0.158363	+	0.987381i	$0.550621\pi$
$110$	0	0
$111$	9.04351	0.858372
$112$	0	0
$113$	−17.5322	−1.64929	−0.824643	−	0.565653i	$-0.808624\pi$
$113$	−17.5322	−1.64929	−0.824643	+	0.565653i	$0.808624\pi$
$114$	0	0
$115$	4.23912	0.395300
$116$	0	0
$117$	−26.0780	−2.41091
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−27.6947	−2.49714
$124$	0	0
$125$	5.82846	0.521313
$126$	0	0
$127$	−0.660190	−0.0585824	−0.0292912	−	0.999571i	$-0.509325\pi$
$127$	−0.660190	−0.0585824	−0.0292912	+	0.999571i	$0.509325\pi$
$128$	0	0
$129$	−38.3984	−3.38079
$130$	0	0
$131$	19.7141	1.72243	0.861214	−	0.508242i	$-0.169704\pi$
$131$	19.7141	1.72243	0.861214	+	0.508242i	$0.169704\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	44.9007	3.86444
$136$	0	0
$137$	2.77455	0.237046	0.118523	−	0.992951i	$-0.462184\pi$
$137$	2.77455	0.237046	0.118523	+	0.992951i	$0.462184\pi$
$138$	0	0
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	13.4887	1.13595
$142$	0	0
$143$	−3.66019	−0.306080
$144$	0	0
$145$	−9.91874	−0.823707
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.79863	−0.720812	−0.360406	−	0.932796i	$-0.617362\pi$
$149$	−8.79863	−0.720812	−0.360406	+	0.932796i	$0.617362\pi$
$150$	0	0
$151$	20.3880	1.65915	0.829574	−	0.558396i	$-0.188583\pi$
$151$	20.3880	1.65915	0.829574	+	0.558396i	$0.188583\pi$
$152$	0	0
$153$	12.5458	1.01427
$154$	0	0
$155$	32.3789	2.60074
$156$	0	0
$157$	7.59261	0.605956	0.302978	−	0.952998i	$-0.402019\pi$
$157$	7.59261	0.605956	0.302978	+	0.952998i	$0.402019\pi$
$158$	0	0
$159$	−0.717370	−0.0568911
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.04351	0.630016	0.315008	−	0.949089i	$-0.397993\pi$
$163$	8.04351	0.630016	0.315008	+	0.949089i	$0.397993\pi$
$164$	0	0
$165$	10.8856	0.847446
$166$	0	0
$167$	15.7414	1.21811	0.609055	−	0.793128i	$-0.291549\pi$
$167$	15.7414	1.21811	0.609055	+	0.793128i	$0.291549\pi$
$168$	0	0
$169$	0.396990	0.0305377
$170$	0	0
$171$	26.7954	2.04909
$172$	0	0
$173$	3.74720	0.284895	0.142447	−	0.989802i	$-0.454503\pi$
$173$	3.74720	0.284895	0.142447	+	0.989802i	$0.454503\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−20.8285	−1.56556
$178$	0	0
$179$	11.5023	0.859724	0.429862	−	0.902895i	$-0.358562\pi$
$179$	11.5023	0.859724	0.429862	+	0.902895i	$0.358562\pi$
$180$	0	0
$181$	5.79071	0.430420	0.215210	−	0.976568i	$-0.430956\pi$
$181$	5.79071	0.430420	0.215210	+	0.976568i	$0.430956\pi$
$182$	0	0
$183$	32.6134	2.41085
$184$	0	0
$185$	9.72313	0.714859
$186$	0	0
$187$	1.76088	0.128768
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.5218	−1.34019	−0.670094	−	0.742277i	$-0.733746\pi$
$191$	−18.5218	−1.34019	−0.670094	+	0.742277i	$0.733746\pi$
$192$	0	0
$193$	−3.77455	−0.271698	−0.135849	−	0.990730i	$-0.543376\pi$
$193$	−3.77455	−0.271698	−0.135849	+	0.990730i	$0.543376\pi$
$194$	0	0
$195$	−39.8435	−2.85325
$196$	0	0
$197$	10.1683	0.724459	0.362230	−	0.932089i	$-0.382016\pi$
$197$	10.1683	0.724459	0.362230	+	0.932089i	$0.382016\pi$
$198$	0	0
$199$	−10.0780	−0.714410	−0.357205	−	0.934026i	$-0.616270\pi$
$199$	−10.0780	−0.714410	−0.357205	+	0.934026i	$0.616270\pi$
$200$	0	0
$201$	10.8421	0.764745
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−29.7759	−2.07964
$206$	0	0
$207$	8.82846	0.613620
$208$	0	0
$209$	3.76088	0.260145
$210$	0	0
$211$	1.72313	0.118625	0.0593125	−	0.998239i	$-0.481109\pi$
$211$	1.72313	0.118625	0.0593125	+	0.998239i	$0.481109\pi$
$212$	0	0
$213$	13.4887	0.924227
$214$	0	0
$215$	−41.2840	−2.81555
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−32.3984	−2.18928
$220$	0	0
$221$	−6.44514	−0.433547
$222$	0	0
$223$	−9.47825	−0.634710	−0.317355	−	0.948307i	$-0.602795\pi$
$223$	−9.47825	−0.634710	−0.317355	+	0.948307i	$0.602795\pi$
$224$	0	0
$225$	47.7623	3.18415
$226$	0	0
$227$	−19.4419	−1.29040	−0.645201	−	0.764013i	$-0.723226\pi$
$227$	−19.4419	−1.29040	−0.645201	+	0.764013i	$0.723226\pi$
$228$	0	0
$229$	10.3204	0.681990	0.340995	−	0.940065i	$-0.389236\pi$
$229$	10.3204	0.681990	0.340995	+	0.940065i	$0.389236\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.11436	0.0730041	0.0365021	−	0.999334i	$-0.488378\pi$
$233$	1.11436	0.0730041	0.0365021	+	0.999334i	$0.488378\pi$
$234$	0	0
$235$	14.5023	0.946027
$236$	0	0
$237$	23.3880	1.51921
$238$	0	0
$239$	26.1053	1.68861	0.844307	−	0.535860i	$-0.180013\pi$
$239$	26.1053	1.68861	0.844307	+	0.535860i	$0.180013\pi$
$240$	0	0
$241$	−14.7954	−0.953053	−0.476526	−	0.879160i	$-0.658104\pi$
$241$	−14.7954	−0.953053	−0.476526	+	0.879160i	$0.658104\pi$
$242$	0	0
$243$	25.4991	1.63577
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−13.7655	−0.875879
$248$	0	0
$249$	−33.3743	−2.11501
$250$	0	0
$251$	−1.96225	−0.123856	−0.0619281	−	0.998081i	$-0.519725\pi$
$251$	−1.96225	−0.123856	−0.0619281	+	0.998081i	$0.519725\pi$
$252$	0	0
$253$	1.23912	0.0779030
$254$	0	0
$255$	19.1683	1.20036
$256$	0	0
$257$	−2.50808	−0.156450	−0.0782249	−	0.996936i	$-0.524925\pi$
$257$	−2.50808	−0.156450	−0.0782249	+	0.996936i	$0.524925\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−20.6569	−1.27863
$262$	0	0
$263$	4.16827	0.257027	0.128513	−	0.991708i	$-0.458980\pi$
$263$	4.16827	0.257027	0.128513	+	0.991708i	$0.458980\pi$
$264$	0	0
$265$	−0.771280	−0.0473794
$266$	0	0
$267$	−23.1111	−1.41438
$268$	0	0
$269$	21.1488	1.28947	0.644734	−	0.764407i	$-0.276968\pi$
$269$	21.1488	1.28947	0.644734	+	0.764407i	$0.276968\pi$
$270$	0	0
$271$	13.2769	0.806513	0.403256	−	0.915087i	$-0.367878\pi$
$271$	13.2769	0.806513	0.403256	+	0.915087i	$0.367878\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.70370	0.404248
$276$	0	0
$277$	−9.50232	−0.570939	−0.285470	−	0.958388i	$-0.592150\pi$
$277$	−9.50232	−0.570939	−0.285470	+	0.958388i	$0.592150\pi$
$278$	0	0
$279$	67.4328	4.03710
$280$	0	0
$281$	26.8766	1.60332	0.801662	−	0.597777i	$-0.203949\pi$
$281$	26.8766	1.60332	0.801662	+	0.597777i	$0.203949\pi$
$282$	0	0
$283$	0.309976	0.0184262	0.00921309	−	0.999958i	$-0.497067\pi$
$283$	0.309976	0.0184262	0.00921309	+	0.999958i	$0.497067\pi$
$284$	0	0
$285$	40.9396	2.42505
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.8993	−0.817607
$290$	0	0
$291$	−37.3789	−2.19119
$292$	0	0
$293$	−31.2222	−1.82402	−0.912010	−	0.410169i	$-0.865470\pi$
$293$	−31.2222	−1.82402	−0.912010	+	0.410169i	$0.865470\pi$
$294$	0	0
$295$	−22.3937	−1.30381
$296$	0	0
$297$	13.1248	0.761576
$298$	0	0
$299$	−4.53543	−0.262290
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−38.3549	−2.20343
$304$	0	0
$305$	35.0643	2.00778
$306$	0	0
$307$	7.50808	0.428509	0.214254	−	0.976778i	$-0.431268\pi$
$307$	7.50808	0.428509	0.214254	+	0.976778i	$0.431268\pi$
$308$	0	0
$309$	−53.7623	−3.05843
$310$	0	0
$311$	−1.53543	−0.0870660	−0.0435330	−	0.999052i	$-0.513861\pi$
$311$	−1.53543	−0.0870660	−0.0435330	+	0.999052i	$0.513861\pi$
$312$	0	0
$313$	12.8525	0.726468	0.363234	−	0.931698i	$-0.381672\pi$
$313$	12.8525	0.726468	0.363234	+	0.931698i	$0.381672\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.43147	0.136565	0.0682825	−	0.997666i	$-0.478248\pi$
$317$	2.43147	0.136565	0.0682825	+	0.997666i	$0.478248\pi$
$318$	0	0
$319$	−2.89931	−0.162330
$320$	0	0
$321$	−3.64076	−0.203207
$322$	0	0
$323$	6.62244	0.368482
$324$	0	0
$325$	−24.5368	−1.36106
$326$	0	0
$327$	−10.5218	−0.581854
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−10.4074	−0.572042	−0.286021	−	0.958223i	$-0.592333\pi$
$331$	−10.4074	−0.572042	−0.286021	+	0.958223i	$0.592333\pi$
$332$	0	0
$333$	20.2495	1.10967
$334$	0	0
$335$	11.6569	0.636886
$336$	0	0
$337$	−11.2164	−0.610998	−0.305499	−	0.952192i	$-0.598823\pi$
$337$	−11.2164	−0.610998	−0.305499	+	0.952192i	$0.598823\pi$
$338$	0	0
$339$	−55.7863	−3.02990
$340$	0	0
$341$	9.46457	0.512535
$342$	0	0
$343$	0	0
$344$	0	0
$345$	13.4887	0.726205
$346$	0	0
$347$	26.7576	1.43642	0.718212	−	0.695825i	$-0.244961\pi$
$347$	26.7576	1.43642	0.718212	+	0.695825i	$0.244961\pi$
$348$	0	0
$349$	5.11436	0.273765	0.136883	−	0.990587i	$-0.456292\pi$
$349$	5.11436	0.273765	0.136883	+	0.990587i	$0.456292\pi$
$350$	0	0
$351$	−48.0391	−2.56414
$352$	0	0
$353$	20.3009	1.08051	0.540255	−	0.841501i	$-0.318328\pi$
$353$	20.3009	1.08051	0.540255	+	0.841501i	$0.318328\pi$
$354$	0	0
$355$	14.5023	0.769703
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.46784	−0.499694	−0.249847	−	0.968285i	$-0.580380\pi$
$359$	−9.46784	−0.499694	−0.249847	+	0.968285i	$0.580380\pi$
$360$	0	0
$361$	−4.85581	−0.255569
$362$	0	0
$363$	3.18194	0.167009
$364$	0	0
$365$	−34.8331	−1.82325
$366$	0	0
$367$	21.4646	1.12044	0.560221	−	0.828343i	$-0.310716\pi$
$367$	21.4646	1.12044	0.560221	+	0.828343i	$0.310716\pi$
$368$	0	0
$369$	−62.0118	−3.22820
$370$	0	0
$371$	0	0
$372$	0	0
$373$	13.3365	0.690540	0.345270	−	0.938503i	$-0.387787\pi$
$373$	13.3365	0.690540	0.345270	+	0.938503i	$0.387787\pi$
$374$	0	0
$375$	18.5458	0.957703
$376$	0	0
$377$	10.6120	0.546548
$378$	0	0
$379$	5.76088	0.295916	0.147958	−	0.988994i	$-0.452730\pi$
$379$	5.76088	0.295916	0.147958	+	0.988994i	$0.452730\pi$
$380$	0	0
$381$	−2.10069	−0.107621
$382$	0	0
$383$	−29.2301	−1.49359	−0.746794	−	0.665055i	$-0.768408\pi$
$383$	−29.2301	−1.49359	−0.746794	+	0.665055i	$0.768408\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−85.9787	−4.37054
$388$	0	0
$389$	−20.5081	−1.03980	−0.519900	−	0.854227i	$-0.674031\pi$
$389$	−20.5081	−1.03980	−0.519900	+	0.854227i	$0.674031\pi$
$390$	0	0
$391$	2.18194	0.110346
$392$	0	0
$393$	62.7292	3.16427
$394$	0	0
$395$	25.1456	1.26521
$396$	0	0
$397$	−11.5699	−0.580677	−0.290338	−	0.956924i	$-0.593768\pi$
$397$	−11.5699	−0.580677	−0.290338	+	0.956924i	$0.593768\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	25.1866	1.25776	0.628879	−	0.777503i	$-0.283514\pi$
$401$	25.1866	1.25776	0.628879	+	0.777503i	$0.283514\pi$
$402$	0	0
$403$	−34.6421	−1.72565
$404$	0	0
$405$	69.7486	3.46583
$406$	0	0
$407$	2.84213	0.140879
$408$	0	0
$409$	−3.80765	−0.188276	−0.0941382	−	0.995559i	$-0.530010\pi$
$409$	−3.80765	−0.188276	−0.0941382	+	0.995559i	$0.530010\pi$
$410$	0	0
$411$	8.82846	0.435476
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−35.8824	−1.76140
$416$	0	0
$417$	35.0014	1.71402
$418$	0	0
$419$	−5.08701	−0.248517	−0.124258	−	0.992250i	$-0.539655\pi$
$419$	−5.08701	−0.248517	−0.124258	+	0.992250i	$0.539655\pi$
$420$	0	0
$421$	−16.9442	−0.825810	−0.412905	−	0.910774i	$-0.635486\pi$
$421$	−16.9442	−0.825810	−0.412905	+	0.910774i	$0.635486\pi$
$422$	0	0
$423$	30.2028	1.46851
$424$	0	0
$425$	11.8044	0.572597
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−11.6465	−0.562299
$430$	0	0
$431$	−2.45555	−0.118280	−0.0591398	−	0.998250i	$-0.518836\pi$
$431$	−2.45555	−0.118280	−0.0591398	+	0.998250i	$0.518836\pi$
$432$	0	0
$433$	32.8629	1.57929	0.789646	−	0.613563i	$-0.210264\pi$
$433$	32.8629	1.57929	0.789646	+	0.613563i	$0.210264\pi$
$434$	0	0
$435$	−31.5609	−1.51323
$436$	0	0
$437$	4.66019	0.222927
$438$	0	0
$439$	−0.0812565	−0.00387816	−0.00193908	−	0.999998i	$-0.500617\pi$
$439$	−0.0812565	−0.00387816	−0.00193908	+	0.999998i	$0.500617\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.73680	0.367586	0.183793	−	0.982965i	$-0.441162\pi$
$443$	7.73680	0.367586	0.183793	+	0.982965i	$0.441162\pi$
$444$	0	0
$445$	−24.8479	−1.17790
$446$	0	0
$447$	−27.9967	−1.32420
$448$	0	0
$449$	−21.1078	−0.996140	−0.498070	−	0.867137i	$-0.665958\pi$
$449$	−21.1078	−0.996140	−0.498070	+	0.867137i	$0.665958\pi$
$450$	0	0
$451$	−8.70370	−0.409841
$452$	0	0
$453$	64.8733	3.04802
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.7174	−0.782006	−0.391003	−	0.920389i	$-0.627872\pi$
$457$	−16.7174	−0.782006	−0.391003	+	0.920389i	$0.627872\pi$
$458$	0	0
$459$	23.1111	1.07873
$460$	0	0
$461$	5.62571	0.262015	0.131008	−	0.991381i	$-0.458179\pi$
$461$	5.62571	0.262015	0.131008	+	0.991381i	$0.458179\pi$
$462$	0	0
$463$	−24.3341	−1.13090	−0.565450	−	0.824783i	$-0.691297\pi$
$463$	−24.3341	−1.13090	−0.565450	+	0.824783i	$0.691297\pi$
$464$	0	0
$465$	103.028	4.77781
$466$	0	0
$467$	−18.8479	−0.872176	−0.436088	−	0.899904i	$-0.643636\pi$
$467$	−18.8479	−0.872176	−0.436088	+	0.899904i	$0.643636\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	24.1592	1.11320
$472$	0	0
$473$	−12.0676	−0.554868
$474$	0	0
$475$	25.2118	1.15680
$476$	0	0
$477$	−1.60628	−0.0735465
$478$	0	0
$479$	29.8766	1.36510	0.682549	−	0.730840i	$-0.260872\pi$
$479$	29.8766	1.36510	0.682549	+	0.730840i	$0.260872\pi$
$480$	0	0
$481$	−10.4027	−0.474324
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−40.1880	−1.82484
$486$	0	0
$487$	12.4315	0.563324	0.281662	−	0.959514i	$-0.409114\pi$
$487$	12.4315	0.563324	0.281662	+	0.959514i	$0.409114\pi$
$488$	0	0
$489$	25.5940	1.15740
$490$	0	0
$491$	−10.0917	−0.455430	−0.227715	−	0.973728i	$-0.573125\pi$
$491$	−10.0917	−0.455430	−0.227715	+	0.973728i	$0.573125\pi$
$492$	0	0
$493$	−5.10533	−0.229933
$494$	0	0
$495$	24.3743	1.09554
$496$	0	0
$497$	0	0
$498$	0	0
$499$	15.4854	0.693221	0.346610	−	0.938009i	$-0.387333\pi$
$499$	15.4854	0.693221	0.346610	+	0.938009i	$0.387333\pi$
$500$	0	0
$501$	50.0884	2.23778
$502$	0	0
$503$	−3.99673	−0.178205	−0.0891027	−	0.996022i	$-0.528400\pi$
$503$	−3.99673	−0.178205	−0.0891027	+	0.996022i	$0.528400\pi$
$504$	0	0
$505$	−41.2372	−1.83503
$506$	0	0
$507$	1.26320	0.0561007
$508$	0	0
$509$	1.78358	0.0790556	0.0395278	−	0.999218i	$-0.487415\pi$
$509$	1.78358	0.0790556	0.0395278	+	0.999218i	$0.487415\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	49.3606	2.17932
$514$	0	0
$515$	−57.8025	−2.54708
$516$	0	0
$517$	4.23912	0.186436
$518$	0	0
$519$	11.9234	0.523379
$520$	0	0
$521$	−33.1729	−1.45333	−0.726666	−	0.686991i	$-0.758931\pi$
$521$	−33.1729	−1.45333	−0.726666	+	0.686991i	$0.758931\pi$
$522$	0	0
$523$	−25.2438	−1.10383	−0.551916	−	0.833899i	$-0.686103\pi$
$523$	−25.2438	−1.10383	−0.551916	+	0.833899i	$0.686103\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.6659	0.725980
$528$	0	0
$529$	−21.4646	−0.933242
$530$	0	0
$531$	−46.6375	−2.02389
$532$	0	0
$533$	31.8572	1.37989
$534$	0	0
$535$	−3.91436	−0.169233
$536$	0	0
$537$	36.5997	1.57940
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−7.94282	−0.341489	−0.170744	−	0.985315i	$-0.554617\pi$
$541$	−7.94282	−0.341489	−0.170744	+	0.985315i	$0.554617\pi$
$542$	0	0
$543$	18.4257	0.790723
$544$	0	0
$545$	−11.3125	−0.484573
$546$	0	0
$547$	−17.2495	−0.737537	−0.368768	−	0.929521i	$-0.620220\pi$
$547$	−17.2495	−0.737537	−0.368768	+	0.929521i	$0.620220\pi$
$548$	0	0
$549$	73.0255	3.11665
$550$	0	0
$551$	−10.9040	−0.464524
$552$	0	0
$553$	0	0
$554$	0	0
$555$	30.9384	1.31326
$556$	0	0
$557$	11.2586	0.477040	0.238520	−	0.971138i	$-0.423338\pi$
$557$	11.2586	0.477040	0.238520	+	0.971138i	$0.423338\pi$
$558$	0	0
$559$	44.1696	1.86818
$560$	0	0
$561$	5.60301	0.236559
$562$	0	0
$563$	−10.3229	−0.435057	−0.217528	−	0.976054i	$-0.569800\pi$
$563$	−10.3229	−0.435057	−0.217528	+	0.976054i	$0.569800\pi$
$564$	0	0
$565$	−59.9787	−2.52332
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.1521	0.467521	0.233760	−	0.972294i	$-0.424897\pi$
$569$	11.1521	0.467521	0.233760	+	0.972294i	$0.424897\pi$
$570$	0	0
$571$	−8.57566	−0.358880	−0.179440	−	0.983769i	$-0.557429\pi$
$571$	−8.57566	−0.358880	−0.179440	+	0.983769i	$0.557429\pi$
$572$	0	0
$573$	−58.9352	−2.46205
$574$	0	0
$575$	8.30671	0.346414
$576$	0	0
$577$	−43.5803	−1.81427	−0.907136	−	0.420838i	$-0.861736\pi$
$577$	−43.5803	−1.81427	−0.907136	+	0.420838i	$0.861736\pi$
$578$	0	0
$579$	−12.0104	−0.499135
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−0.225450	−0.00933719
$584$	0	0
$585$	−89.2145	−3.68857
$586$	0	0
$587$	−44.4315	−1.83388	−0.916942	−	0.399022i	$-0.869350\pi$
$587$	−44.4315	−1.83388	−0.916942	+	0.399022i	$0.869350\pi$
$588$	0	0
$589$	35.5951	1.46667
$590$	0	0
$591$	32.3549	1.33090
$592$	0	0
$593$	5.03448	0.206741	0.103371	−	0.994643i	$-0.467037\pi$
$593$	5.03448	0.206741	0.103371	+	0.994643i	$0.467037\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−32.0676	−1.31244
$598$	0	0
$599$	6.73569	0.275213	0.137606	−	0.990487i	$-0.456059\pi$
$599$	6.73569	0.275213	0.137606	+	0.990487i	$0.456059\pi$
$600$	0	0
$601$	30.4854	1.24352	0.621762	−	0.783206i	$-0.286417\pi$
$601$	30.4854	1.24352	0.621762	+	0.783206i	$0.286417\pi$
$602$	0	0
$603$	24.2769	0.988631
$604$	0	0
$605$	3.42107	0.139086
$606$	0	0
$607$	27.3171	1.10877	0.554384	−	0.832261i	$-0.312954\pi$
$607$	27.3171	1.10877	0.554384	+	0.832261i	$0.312954\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.5160	−0.627710
$612$	0	0
$613$	−34.2495	−1.38333	−0.691663	−	0.722221i	$-0.743121\pi$
$613$	−34.2495	−1.38333	−0.691663	+	0.722221i	$0.743121\pi$
$614$	0	0
$615$	−94.7453	−3.82050
$616$	0	0
$617$	9.13052	0.367581	0.183790	−	0.982965i	$-0.441163\pi$
$617$	9.13052	0.367581	0.183790	+	0.982965i	$0.441163\pi$
$618$	0	0
$619$	−7.32038	−0.294231	−0.147115	−	0.989119i	$-0.546999\pi$
$619$	−7.32038	−0.294231	−0.147115	+	0.989119i	$0.546999\pi$
$620$	0	0
$621$	16.2632	0.652620
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−13.5789	−0.543157
$626$	0	0
$627$	11.9669	0.477912
$628$	0	0
$629$	5.00465	0.199548
$630$	0	0
$631$	−12.7199	−0.506370	−0.253185	−	0.967418i	$-0.581478\pi$
$631$	−12.7199	−0.506370	−0.253185	+	0.967418i	$0.581478\pi$
$632$	0	0
$633$	5.48289	0.217925
$634$	0	0
$635$	−2.25855	−0.0896280
$636$	0	0
$637$	0	0
$638$	0	0
$639$	30.2028	1.19480
$640$	0	0
$641$	−8.16500	−0.322498	−0.161249	−	0.986914i	$-0.551552\pi$
$641$	−8.16500	−0.322498	−0.161249	+	0.986914i	$0.551552\pi$
$642$	0	0
$643$	8.39123	0.330918	0.165459	−	0.986217i	$-0.447089\pi$
$643$	8.39123	0.330918	0.165459	+	0.986217i	$0.447089\pi$
$644$	0	0
$645$	−131.363	−5.17243
$646$	0	0
$647$	40.5199	1.59300	0.796500	−	0.604638i	$-0.206682\pi$
$647$	40.5199	1.59300	0.796500	+	0.604638i	$0.206682\pi$
$648$	0	0
$649$	−6.54583	−0.256946
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.7037	0.575400	0.287700	−	0.957721i	$-0.407109\pi$
$653$	14.7037	0.575400	0.287700	+	0.957721i	$0.407109\pi$
$654$	0	0
$655$	67.4433	2.63523
$656$	0	0
$657$	−72.5439	−2.83021
$658$	0	0
$659$	−19.2873	−0.751326	−0.375663	−	0.926756i	$-0.622585\pi$
$659$	−19.2873	−0.751326	−0.375663	+	0.926756i	$0.622585\pi$
$660$	0	0
$661$	19.8123	0.770609	0.385305	−	0.922789i	$-0.374096\pi$
$661$	19.8123	0.770609	0.385305	+	0.922789i	$0.374096\pi$
$662$	0	0
$663$	−20.5081	−0.796468
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.59261	−0.139106
$668$	0	0
$669$	−30.1592	−1.16602
$670$	0	0
$671$	10.2495	0.395679
$672$	0	0
$673$	−1.11901	−0.0431345	−0.0215673	−	0.999767i	$-0.506866\pi$
$673$	−1.11901	−0.0431345	−0.0215673	+	0.999767i	$0.506866\pi$
$674$	0	0
$675$	87.9844	3.38652
$676$	0	0
$677$	−43.5141	−1.67238	−0.836191	−	0.548438i	$-0.815223\pi$
$677$	−43.5141	−1.67238	−0.836191	+	0.548438i	$0.815223\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−61.8629	−2.37059
$682$	0	0
$683$	15.0949	0.577591	0.288796	−	0.957391i	$-0.406745\pi$
$683$	15.0949	0.577591	0.288796	+	0.957391i	$0.406745\pi$
$684$	0	0
$685$	9.49192	0.362668
$686$	0	0
$687$	32.8389	1.25288
$688$	0	0
$689$	0.825191	0.0314373
$690$	0	0
$691$	2.14171	0.0814743	0.0407372	−	0.999170i	$-0.487029\pi$
$691$	2.14171	0.0814743	0.0407372	+	0.999170i	$0.487029\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	37.6317	1.42745
$696$	0	0
$697$	−15.3261	−0.580519
$698$	0	0
$699$	3.54583	0.134116
$700$	0	0
$701$	−10.1910	−0.384908	−0.192454	−	0.981306i	$-0.561645\pi$
$701$	−10.1910	−0.384908	−0.192454	+	0.981306i	$0.561645\pi$
$702$	0	0
$703$	10.6889	0.403140
$704$	0	0
$705$	46.1456	1.73794
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0.345567	0.0129781	0.00648903	−	0.999979i	$-0.497934\pi$
$709$	0.345567	0.0129781	0.00648903	+	0.999979i	$0.497934\pi$
$710$	0	0
$711$	52.3685	1.96397
$712$	0	0
$713$	11.7278	0.439209
$714$	0	0
$715$	−12.5218	−0.468287
$716$	0	0
$717$	83.0657	3.10215
$718$	0	0
$719$	−26.8766	−1.00233	−0.501164	−	0.865352i	$-0.667095\pi$
$719$	−26.8766	−1.00233	−0.501164	+	0.865352i	$0.667095\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−47.0780	−1.75085
$724$	0	0
$725$	−19.4361	−0.721839
$726$	0	0
$727$	−46.1261	−1.71072	−0.855362	−	0.518031i	$-0.826665\pi$
$727$	−46.1261	−1.71072	−0.855362	+	0.518031i	$0.826665\pi$
$728$	0	0
$729$	19.9727	0.739728
$730$	0	0
$731$	−21.2495	−0.785942
$732$	0	0
$733$	−30.9396	−1.14278	−0.571389	−	0.820679i	$-0.693595\pi$
$733$	−30.9396	−1.14278	−0.571389	+	0.820679i	$0.693595\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.40739	0.125513
$738$	0	0
$739$	−40.2301	−1.47989	−0.739944	−	0.672668i	$-0.765148\pi$
$739$	−40.2301	−1.47989	−0.739944	+	0.672668i	$0.765148\pi$
$740$	0	0
$741$	−43.8011	−1.60907
$742$	0	0
$743$	−37.3710	−1.37101	−0.685505	−	0.728068i	$-0.740418\pi$
$743$	−37.3710	−1.37101	−0.685505	+	0.728068i	$0.740418\pi$
$744$	0	0
$745$	−30.1007	−1.10280
$746$	0	0
$747$	−74.7292	−2.73420
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−39.7396	−1.45012	−0.725058	−	0.688687i	$-0.758187\pi$
$751$	−39.7396	−1.45012	−0.725058	+	0.688687i	$0.758187\pi$
$752$	0	0
$753$	−6.24377	−0.227536
$754$	0	0
$755$	69.7486	2.53841
$756$	0	0
$757$	6.60628	0.240109	0.120055	−	0.992767i	$-0.461693\pi$
$757$	6.60628	0.240109	0.120055	+	0.992767i	$0.461693\pi$
$758$	0	0
$759$	3.94282	0.143115
$760$	0	0
$761$	−1.03775	−0.0376184	−0.0188092	−	0.999823i	$-0.505988\pi$
$761$	−1.03775	−0.0376184	−0.0188092	+	0.999823i	$0.505988\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	42.9201	1.55178
$766$	0	0
$767$	23.9590	0.865109
$768$	0	0
$769$	−6.03199	−0.217519	−0.108760	−	0.994068i	$-0.534688\pi$
$769$	−6.03199	−0.217519	−0.108760	+	0.994068i	$0.534688\pi$
$770$	0	0
$771$	−7.98057	−0.287413
$772$	0	0
$773$	−14.9007	−0.535941	−0.267970	−	0.963427i	$-0.586353\pi$
$773$	−14.9007	−0.535941	−0.267970	+	0.963427i	$0.586353\pi$
$774$	0	0
$775$	63.4476	2.27911
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−32.7335	−1.17280
$780$	0	0
$781$	4.23912	0.151688
$782$	0	0
$783$	−38.0528	−1.35990
$784$	0	0
$785$	25.9748	0.927081
$786$	0	0
$787$	0.980570	0.0349535	0.0174768	−	0.999847i	$-0.494437\pi$
$787$	0.980570	0.0349535	0.0174768	+	0.999847i	$0.494437\pi$
$788$	0	0
$789$	13.2632	0.472182
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−37.5152	−1.33220
$794$	0	0
$795$	−2.45417	−0.0870404
$796$	0	0
$797$	37.2873	1.32078	0.660392	−	0.750921i	$-0.270390\pi$
$797$	37.2873	1.32078	0.660392	+	0.750921i	$0.270390\pi$
$798$	0	0
$799$	7.46457	0.264078
$800$	0	0
$801$	−51.7486	−1.82845
$802$	0	0
$803$	−10.1819	−0.359313
$804$	0	0
$805$	0	0
$806$	0	0
$807$	67.2944	2.36888
$808$	0	0
$809$	26.8468	0.943883	0.471941	−	0.881630i	$-0.343553\pi$
$809$	26.8468	0.943883	0.471941	+	0.881630i	$0.343553\pi$
$810$	0	0
$811$	31.0572	1.09057	0.545283	−	0.838252i	$-0.316422\pi$
$811$	31.0572	1.09057	0.545283	+	0.838252i	$0.316422\pi$
$812$	0	0
$813$	42.2463	1.48164
$814$	0	0
$815$	27.5174	0.963892
$816$	0	0
$817$	−45.3847	−1.58781
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19.8446	−0.692582	−0.346291	−	0.938127i	$-0.612559\pi$
$821$	−19.8446	−0.692582	−0.346291	+	0.938127i	$0.612559\pi$
$822$	0	0
$823$	−18.5836	−0.647783	−0.323891	−	0.946094i	$-0.604991\pi$
$823$	−18.5836	−0.647783	−0.323891	+	0.946094i	$0.604991\pi$
$824$	0	0
$825$	21.3308	0.742643
$826$	0	0
$827$	−7.25744	−0.252366	−0.126183	−	0.992007i	$-0.540273\pi$
$827$	−7.25744	−0.252366	−0.126183	+	0.992007i	$0.540273\pi$
$828$	0	0
$829$	42.9787	1.49271	0.746356	−	0.665547i	$-0.231802\pi$
$829$	42.9787	1.49271	0.746356	+	0.665547i	$0.231802\pi$
$830$	0	0
$831$	−30.2359	−1.04887
$832$	0	0
$833$	0	0
$834$	0	0
$835$	53.8525	1.86364
$836$	0	0
$837$	124.220	4.29368
$838$	0	0
$839$	35.1650	1.21403	0.607015	−	0.794690i	$-0.292367\pi$
$839$	35.1650	1.21403	0.607015	+	0.794690i	$0.292367\pi$
$840$	0	0
$841$	−20.5940	−0.710137
$842$	0	0
$843$	85.5199	2.94546
$844$	0	0
$845$	1.35813	0.0467211
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0.986327	0.0338507
$850$	0	0
$851$	3.52175	0.120724
$852$	0	0
$853$	40.5587	1.38870	0.694352	−	0.719635i	$-0.255691\pi$
$853$	40.5587	1.38870	0.694352	+	0.719635i	$0.255691\pi$
$854$	0	0
$855$	91.6687	3.13500
$856$	0	0
$857$	42.1956	1.44137	0.720687	−	0.693260i	$-0.243826\pi$
$857$	42.1956	1.44137	0.720687	+	0.693260i	$0.243826\pi$
$858$	0	0
$859$	53.1431	1.81322	0.906609	−	0.421971i	$-0.138662\pi$
$859$	53.1431	1.81322	0.906609	+	0.421971i	$0.138662\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31.9863	−1.08883	−0.544414	−	0.838817i	$-0.683248\pi$
$863$	−31.9863	−1.08883	−0.544414	+	0.838817i	$0.683248\pi$
$864$	0	0
$865$	12.8194	0.435874
$866$	0	0
$867$	−44.2268	−1.50202
$868$	0	0
$869$	7.35021	0.249339
$870$	0	0
$871$	−12.4717	−0.422588
$872$	0	0
$873$	−83.6960	−2.83268
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.6922	0.664958	0.332479	−	0.943111i	$-0.392115\pi$
$877$	19.6922	0.664958	0.332479	+	0.943111i	$0.392115\pi$
$878$	0	0
$879$	−99.3472	−3.35090
$880$	0	0
$881$	42.3333	1.42624	0.713122	−	0.701040i	$-0.247281\pi$
$881$	42.3333	1.42624	0.713122	+	0.701040i	$0.247281\pi$
$882$	0	0
$883$	−22.5023	−0.757263	−0.378632	−	0.925547i	$-0.623605\pi$
$883$	−22.5023	−0.757263	−0.378632	+	0.925547i	$0.623605\pi$
$884$	0	0
$885$	−71.2555	−2.39523
$886$	0	0
$887$	−25.8572	−0.868199	−0.434100	−	0.900865i	$-0.642933\pi$
$887$	−25.8572	−0.868199	−0.434100	+	0.900865i	$0.642933\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	20.3880	0.683022
$892$	0	0
$893$	15.9428	0.533506
$894$	0	0
$895$	39.3502	1.31533
$896$	0	0
$897$	−14.4315	−0.481853
$898$	0	0
$899$	−27.4408	−0.915201
$900$	0	0
$901$	−0.396990	−0.0132257
$902$	0	0
$903$	0	0
$904$	0	0
$905$	19.8104	0.658520
$906$	0	0
$907$	−14.5814	−0.484168	−0.242084	−	0.970255i	$-0.577831\pi$
$907$	−14.5814	−0.484168	−0.242084	+	0.970255i	$0.577831\pi$
$908$	0	0
$909$	−85.8813	−2.84850
$910$	0	0
$911$	8.63860	0.286210	0.143105	−	0.989708i	$-0.454291\pi$
$911$	8.63860	0.286210	0.143105	+	0.989708i	$0.454291\pi$
$912$	0	0
$913$	−10.4887	−0.347124
$914$	0	0
$915$	111.573	3.68848
$916$	0	0
$917$	0	0
$918$	0	0
$919$	31.0482	1.02418	0.512092	−	0.858931i	$-0.328871\pi$
$919$	31.0482	1.02418	0.512092	+	0.858931i	$0.328871\pi$
$920$	0	0
$921$	23.8903	0.787212
$922$	0	0
$923$	−15.5160	−0.510715
$924$	0	0
$925$	19.0528	0.626452
$926$	0	0
$927$	−120.380	−3.95381
$928$	0	0
$929$	−7.92666	−0.260065	−0.130033	−	0.991510i	$-0.541508\pi$
$929$	−7.92666	−0.260065	−0.130033	+	0.991510i	$0.541508\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−4.88564	−0.159949
$934$	0	0
$935$	6.02408	0.197008
$936$	0	0
$937$	9.42571	0.307925	0.153962	−	0.988077i	$-0.450797\pi$
$937$	9.42571	0.307925	0.153962	+	0.988077i	$0.450797\pi$
$938$	0	0
$939$	40.8960	1.33459
$940$	0	0
$941$	46.3653	1.51146	0.755732	−	0.654881i	$-0.227281\pi$
$941$	46.3653	1.51146	0.755732	+	0.654881i	$0.227281\pi$
$942$	0	0
$943$	−10.7850	−0.351206
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.12149	−0.133931	−0.0669653	−	0.997755i	$-0.521332\pi$
$947$	−4.12149	−0.133931	−0.0669653	+	0.997755i	$0.521332\pi$
$948$	0	0
$949$	37.2678	1.20977
$950$	0	0
$951$	7.73680	0.250883
$952$	0	0
$953$	37.6446	1.21943	0.609714	−	0.792621i	$-0.291284\pi$
$953$	37.6446	1.21943	0.609714	+	0.792621i	$0.291284\pi$
$954$	0	0
$955$	−63.3642	−2.05042
$956$	0	0
$957$	−9.22545	−0.298216
$958$	0	0
$959$	0	0
$960$	0	0
$961$	58.5782	1.88962
$962$	0	0
$963$	−8.15211	−0.262698
$964$	0	0
$965$	−12.9130	−0.415684
$966$	0	0
$967$	1.22545	0.0394078	0.0197039	−	0.999806i	$-0.493728\pi$
$967$	1.22545	0.0394078	0.0197039	+	0.999806i	$0.493728\pi$
$968$	0	0
$969$	21.0722	0.676938
$970$	0	0
$971$	−33.3628	−1.07066	−0.535331	−	0.844642i	$-0.679813\pi$
$971$	−33.3628	−1.07066	−0.535331	+	0.844642i	$0.679813\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−78.0747	−2.50039
$976$	0	0
$977$	36.2919	1.16108	0.580541	−	0.814231i	$-0.302841\pi$
$977$	36.2919	1.16108	0.580541	+	0.814231i	$0.302841\pi$
$978$	0	0
$979$	−7.26320	−0.232133
$980$	0	0
$981$	−23.5595	−0.752197
$982$	0	0
$983$	−1.14995	−0.0366777	−0.0183389	−	0.999832i	$-0.505838\pi$
$983$	−1.14995	−0.0366777	−0.0183389	+	0.999832i	$0.505838\pi$
$984$	0	0
$985$	34.7863	1.10838
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−14.9532	−0.475485
$990$	0	0
$991$	−25.5206	−0.810690	−0.405345	−	0.914164i	$-0.632849\pi$
$991$	−25.5206	−0.810690	−0.405345	+	0.914164i	$0.632849\pi$
$992$	0	0
$993$	−33.1157	−1.05090
$994$	0	0
$995$	−34.4775	−1.09301
$996$	0	0
$997$	8.54118	0.270502	0.135251	−	0.990811i	$-0.456816\pi$
$997$	8.54118	0.270502	0.135251	+	0.990811i	$0.456816\pi$
$998$	0	0
$999$	37.3023	1.18019

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.cn.1.3		3
4.3	odd	2		2156.2.a.h.1.1		3
7.3	odd	6		1232.2.q.l.177.3		6
7.5	odd	6		1232.2.q.l.529.3		6
7.6	odd	2		8624.2.a.ci.1.1		3
28.3	even	6		308.2.i.a.177.1	✓	6
28.11	odd	6		2156.2.i.l.177.3		6
28.19	even	6		308.2.i.a.221.1	yes	6
28.23	odd	6		2156.2.i.l.1145.3		6
28.27	even	2		2156.2.a.i.1.3		3
84.47	odd	6		2772.2.s.f.2377.1		6
84.59	odd	6		2772.2.s.f.793.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.2.i.a.177.1	✓	6	28.3	even	6
308.2.i.a.221.1	yes	6	28.19	even	6
1232.2.q.l.177.3		6	7.3	odd	6
1232.2.q.l.529.3		6	7.5	odd	6
2156.2.a.h.1.1		3	4.3	odd	2
2156.2.a.i.1.3		3	28.27	even	2
2156.2.i.l.177.3		6	28.11	odd	6
2156.2.i.l.1145.3		6	28.23	odd	6
2772.2.s.f.793.1		6	84.59	odd	6
2772.2.s.f.2377.1		6	84.47	odd	6
8624.2.a.ci.1.1		3	7.6	odd	2
8624.2.a.cn.1.3		3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+3.18194 q^{3} +3.42107 q^{5} +7.12476 q^{9} +1.00000 q^{11} -3.66019 q^{13} +10.8856 q^{15} +1.76088 q^{17} +3.76088 q^{19} +1.23912 q^{23} +6.70370 q^{25} +13.1248 q^{27} -2.89931 q^{29} +9.46457 q^{31} +3.18194 q^{33} +2.84213 q^{37} -11.6465 q^{39} -8.70370 q^{41} -12.0676 q^{43} +24.3743 q^{45} +4.23912 q^{47} +5.60301 q^{51} -0.225450 q^{53} +3.42107 q^{55} +11.9669 q^{57} -6.54583 q^{59} +10.2495 q^{61} -12.5218 q^{65} +3.40739 q^{67} +3.94282 q^{69} +4.23912 q^{71} -10.1819 q^{73} +21.3308 q^{75} +7.35021 q^{79} +20.3880 q^{81} -10.4887 q^{83} +6.02408 q^{85} -9.22545 q^{87} -7.26320 q^{89} +30.1157 q^{93} +12.8662 q^{95} -11.7472 q^{97} +7.12476 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 2 q^{5} + 4 q^{9} + 3 q^{11} - 3 q^{13} + 15 q^{15} + 5 q^{17} + 11 q^{19} + 4 q^{23} + 11 q^{25} + 22 q^{27} - q^{29} + 19 q^{31} + q^{33} - 8 q^{37} - 17 q^{39} - 17 q^{41} - 10 q^{43}+ \cdots + 4 q^{99}+O(q^{100}) \) 3 * q + q^3 + 2 * q^5 + 4 * q^9 + 3 * q^11 - 3 * q^13 + 15 * q^15 + 5 * q^17 + 11 * q^19 + 4 * q^23 + 11 * q^25 + 22 * q^27 - q^29 + 19 * q^31 + q^33 - 8 * q^37 - 17 * q^39 - 17 * q^41 - 10 * q^43 + 21 * q^45 + 13 * q^47 + 9 * q^53 + 2 * q^55 + 2 * q^57 + 6 * q^59 - 4 * q^61 - 37 * q^65 - 8 * q^67 + 3 * q^69 + 13 * q^71 - 22 * q^73 + 13 * q^75 - 5 * q^79 + 19 * q^81 + 3 * q^83 - 7 * q^85 - 18 * q^87 + 3 * q^89 + 14 * q^93 - 3 * q^95 - 25 * q^97 + 4 * q^99

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