LMFDB - Embedded newform 9200.2.a.ct.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$73.4623698596$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.521397.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} - 3x^{2} + 18x + 12 $ x^5 - 9x^3 - 3x^2 + 18*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4600)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.83957$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.83957 q^{3} +3.97272 q^{7} +0.384010 q^{9} +O(q^{10})$	$q-1.83957 q^{3} +3.97272 q^{7} +0.384010 q^{9} -2.10339 q^{11} +5.35673 q^{13} +1.29567 q^{17} -2.10339 q^{19} -7.30809 q^{21} +1.00000 q^{23} +4.81229 q^{27} +6.03130 q^{29} +8.32489 q^{31} +3.86933 q^{33} +5.10131 q^{37} -9.85407 q^{39} -8.33994 q^{41} -7.78253 q^{43} -11.3964 q^{47} +8.78253 q^{49} -2.38347 q^{51} -0.573664 q^{53} +3.86933 q^{57} +9.17951 q^{59} +13.5149 q^{61} +1.52557 q^{63} +15.8314 q^{67} -1.83957 q^{69} -14.9449 q^{71} +8.36459 q^{73} -8.35619 q^{77} +9.41149 q^{79} -10.0046 q^{81} -1.51206 q^{83} -11.0950 q^{87} +10.5903 q^{89} +21.2808 q^{91} -15.3142 q^{93} -0.337451 q^{97} -0.807724 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{7} + 3 q^{9}+O(q^{10}) $ 5 * q - 4 * q^7 + 3 * q^9	$ 5 q - 4 q^{7} + 3 q^{9} + 4 q^{13} + 6 q^{17} + 5 q^{23} - 9 q^{27} + 12 q^{29} + 18 q^{31} + 6 q^{33} + 10 q^{37} - 9 q^{39} - 6 q^{41} - 10 q^{43} - 22 q^{47} + 15 q^{49} + 6 q^{51} + 10 q^{53} + 6 q^{57} + q^{59} + 10 q^{61} - 8 q^{67} - 8 q^{71} + 6 q^{73} - 27 q^{81} + 2 q^{83} - 39 q^{87} + 14 q^{89} + 46 q^{91} - 3 q^{93} + 6 q^{97} + 6 q^{99}+O(q^{100}) $ 5 * q - 4 * q^7 + 3 * q^9 + 4 * q^13 + 6 * q^17 + 5 * q^23 - 9 * q^27 + 12 * q^29 + 18 * q^31 + 6 * q^33 + 10 * q^37 - 9 * q^39 - 6 * q^41 - 10 * q^43 - 22 * q^47 + 15 * q^49 + 6 * q^51 + 10 * q^53 + 6 * q^57 + q^59 + 10 * q^61 - 8 * q^67 - 8 * q^71 + 6 * q^73 - 27 * q^81 + 2 * q^83 - 39 * q^87 + 14 * q^89 + 46 * q^91 - 3 * q^93 + 6 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.83957	−1.06208	−0.531038	−	0.847348i	$-0.678198\pi$
$3$	−1.83957	−1.06208	−0.531038	+	0.847348i	$0.678198\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.97272	1.50155	0.750774	−	0.660559i	$-0.229681\pi$
$7$	3.97272	1.50155	0.750774	+	0.660559i	$0.229681\pi$
$8$	0	0
$9$	0.384010	0.128003
$10$	0	0
$11$	−2.10339	−0.634196	−0.317098	−	0.948393i	$-0.602708\pi$
$11$	−2.10339	−0.634196	−0.317098	+	0.948393i	$0.602708\pi$
$12$	0	0
$13$	5.35673	1.48569	0.742845	−	0.669463i	$-0.233476\pi$
$13$	5.35673	1.48569	0.742845	+	0.669463i	$0.233476\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.29567	0.314246	0.157123	−	0.987579i	$-0.449778\pi$
$17$	1.29567	0.314246	0.157123	+	0.987579i	$0.449778\pi$
$18$	0	0
$19$	−2.10339	−0.482551	−0.241276	−	0.970457i	$-0.577566\pi$
$19$	−2.10339	−0.482551	−0.241276	+	0.970457i	$0.577566\pi$
$20$	0	0
$21$	−7.30809	−1.59476
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.81229	0.926126
$28$	0	0
$29$	6.03130	1.11998	0.559992	−	0.828498i	$-0.310804\pi$
$29$	6.03130	1.11998	0.559992	+	0.828498i	$0.310804\pi$
$30$	0	0
$31$	8.32489	1.49519	0.747597	−	0.664153i	$-0.231207\pi$
$31$	8.32489	1.49519	0.747597	+	0.664153i	$0.231207\pi$
$32$	0	0
$33$	3.86933	0.673564
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.10131	0.838650	0.419325	−	0.907836i	$-0.362267\pi$
$37$	5.10131	0.838650	0.419325	+	0.907836i	$0.362267\pi$
$38$	0	0
$39$	−9.85407	−1.57791
$40$	0	0
$41$	−8.33994	−1.30248	−0.651240	−	0.758872i	$-0.725751\pi$
$41$	−8.33994	−1.30248	−0.651240	+	0.758872i	$0.725751\pi$
$42$	0	0
$43$	−7.78253	−1.18682	−0.593412	−	0.804899i	$-0.702220\pi$
$43$	−7.78253	−1.18682	−0.593412	+	0.804899i	$0.702220\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.3964	−1.66234	−0.831171	−	0.556018i	$-0.812329\pi$
$47$	−11.3964	−1.66234	−0.831171	+	0.556018i	$0.812329\pi$
$48$	0	0
$49$	8.78253	1.25465
$50$	0	0
$51$	−2.38347	−0.333752
$52$	0	0
$53$	−0.573664	−0.0787988	−0.0393994	−	0.999224i	$-0.512544\pi$
$53$	−0.573664	−0.0787988	−0.0393994	+	0.999224i	$0.512544\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.86933	0.512505
$58$	0	0
$59$	9.17951	1.19507	0.597535	−	0.801843i	$-0.296147\pi$
$59$	9.17951	1.19507	0.597535	+	0.801843i	$0.296147\pi$
$60$	0	0
$61$	13.5149	1.73040	0.865201	−	0.501425i	$-0.167191\pi$
$61$	13.5149	1.73040	0.865201	+	0.501425i	$0.167191\pi$
$62$	0	0
$63$	1.52557	0.192203
$64$	0	0
$65$	0	0
$66$	0	0
$67$	15.8314	1.93411	0.967055	−	0.254569i	$-0.0819337\pi$
$67$	15.8314	1.93411	0.967055	+	0.254569i	$0.0819337\pi$
$68$	0	0
$69$	−1.83957	−0.221458
$70$	0	0
$71$	−14.9449	−1.77363	−0.886817	−	0.462121i	$-0.847089\pi$
$71$	−14.9449	−1.77363	−0.886817	+	0.462121i	$0.847089\pi$
$72$	0	0
$73$	8.36459	0.979002	0.489501	−	0.872003i	$-0.337179\pi$
$73$	8.36459	0.979002	0.489501	+	0.872003i	$0.337179\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.35619	−0.952276
$78$	0	0
$79$	9.41149	1.05887	0.529437	−	0.848349i	$-0.322403\pi$
$79$	9.41149	1.05887	0.529437	+	0.848349i	$0.322403\pi$
$80$	0	0
$81$	−10.0046	−1.11162
$82$	0	0
$83$	−1.51206	−0.165970	−0.0829849	−	0.996551i	$-0.526445\pi$
$83$	−1.51206	−0.165970	−0.0829849	+	0.996551i	$0.526445\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−11.0950	−1.18951
$88$	0	0
$89$	10.5903	1.12256	0.561282	−	0.827624i	$-0.310308\pi$
$89$	10.5903	1.12256	0.561282	+	0.827624i	$0.310308\pi$
$90$	0	0
$91$	21.2808	2.23084
$92$	0	0
$93$	−15.3142	−1.58801
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.337451	−0.0342630	−0.0171315	−	0.999853i	$-0.505453\pi$
$97$	−0.337451	−0.0342630	−0.0171315	+	0.999853i	$0.505453\pi$
$98$	0	0
$99$	−0.807724	−0.0811793
$100$	0	0
$101$	−6.19119	−0.616047	−0.308023	−	0.951379i	$-0.599667\pi$
$101$	−6.19119	−0.616047	−0.308023	+	0.951379i	$0.599667\pi$
$102$	0	0
$103$	−12.6267	−1.24414	−0.622071	−	0.782961i	$-0.713708\pi$
$103$	−12.6267	−1.24414	−0.622071	+	0.782961i	$0.713708\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.7940	−1.23684	−0.618419	−	0.785848i	$-0.712227\pi$
$107$	−12.7940	−1.23684	−0.618419	+	0.785848i	$0.712227\pi$
$108$	0	0
$109$	14.3165	1.37127	0.685635	−	0.727945i	$-0.259524\pi$
$109$	14.3165	1.37127	0.685635	+	0.727945i	$0.259524\pi$
$110$	0	0
$111$	−9.38421	−0.890710
$112$	0	0
$113$	4.59463	0.432226	0.216113	−	0.976368i	$-0.430662\pi$
$113$	4.59463	0.432226	0.216113	+	0.976368i	$0.430662\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.05704	0.190173
$118$	0	0
$119$	5.14733	0.471855
$120$	0	0
$121$	−6.57574	−0.597795
$122$	0	0
$123$	15.3419	1.38333
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.6623	−1.12360	−0.561801	−	0.827273i	$-0.689891\pi$
$127$	−12.6623	−1.12360	−0.561801	+	0.827273i	$0.689891\pi$
$128$	0	0
$129$	14.3165	1.26050
$130$	0	0
$131$	−4.68370	−0.409217	−0.204609	−	0.978844i	$-0.565592\pi$
$131$	−4.68370	−0.409217	−0.204609	+	0.978844i	$0.565592\pi$
$132$	0	0
$133$	−8.35619	−0.724574
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.468120	−0.0399942	−0.0199971	−	0.999800i	$-0.506366\pi$
$137$	−0.468120	−0.0399942	−0.0199971	+	0.999800i	$0.506366\pi$
$138$	0	0
$139$	−0.739205	−0.0626985	−0.0313493	−	0.999508i	$-0.509980\pi$
$139$	−0.739205	−0.0626985	−0.0313493	+	0.999508i	$0.509980\pi$
$140$	0	0
$141$	20.9645	1.76553
$142$	0	0
$143$	−11.2673	−0.942220
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−16.1561	−1.33253
$148$	0	0
$149$	14.5903	1.19528	0.597640	−	0.801765i	$-0.296105\pi$
$149$	14.5903	1.19528	0.597640	+	0.801765i	$0.296105\pi$
$150$	0	0
$151$	−21.6885	−1.76498	−0.882491	−	0.470329i	$-0.844135\pi$
$151$	−21.6885	−1.76498	−0.882491	+	0.470329i	$0.844135\pi$
$152$	0	0
$153$	0.497550	0.0402245
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.71253	0.615527	0.307763	−	0.951463i	$-0.400420\pi$
$157$	7.71253	0.615527	0.307763	+	0.951463i	$0.400420\pi$
$158$	0	0
$159$	1.05529	0.0836902
$160$	0	0
$161$	3.97272	0.313094
$162$	0	0
$163$	4.69177	0.367488	0.183744	−	0.982974i	$-0.441178\pi$
$163$	4.69177	0.367488	0.183744	+	0.982974i	$0.441178\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.6738	−0.903343	−0.451671	−	0.892184i	$-0.649172\pi$
$167$	−11.6738	−0.903343	−0.451671	+	0.892184i	$0.649172\pi$
$168$	0	0
$169$	15.6946	1.20728
$170$	0	0
$171$	−0.807724	−0.0617682
$172$	0	0
$173$	−8.31723	−0.632347	−0.316174	−	0.948701i	$-0.602398\pi$
$173$	−8.31723	−0.632347	−0.316174	+	0.948701i	$0.602398\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−16.8863	−1.26925
$178$	0	0
$179$	4.07517	0.304592	0.152296	−	0.988335i	$-0.451333\pi$
$179$	4.07517	0.304592	0.152296	+	0.988335i	$0.451333\pi$
$180$	0	0
$181$	2.52349	0.187569	0.0937846	−	0.995593i	$-0.470103\pi$
$181$	2.52349	0.187569	0.0937846	+	0.995593i	$0.470103\pi$
$182$	0	0
$183$	−24.8615	−1.83782
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.72530	−0.199293
$188$	0	0
$189$	19.1179	1.39062
$190$	0	0
$191$	23.9380	1.73210	0.866048	−	0.499961i	$-0.166652\pi$
$191$	23.9380	1.73210	0.866048	+	0.499961i	$0.166652\pi$
$192$	0	0
$193$	18.7733	1.35133	0.675667	−	0.737207i	$-0.263856\pi$
$193$	18.7733	1.35133	0.675667	+	0.737207i	$0.263856\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.9052	1.13320	0.566600	−	0.823993i	$-0.308259\pi$
$197$	15.9052	1.13320	0.566600	+	0.823993i	$0.308259\pi$
$198$	0	0
$199$	−9.87544	−0.700052	−0.350026	−	0.936740i	$-0.613827\pi$
$199$	−9.87544	−0.700052	−0.350026	+	0.936740i	$0.613827\pi$
$200$	0	0
$201$	−29.1229	−2.05417
$202$	0	0
$203$	23.9607	1.68171
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.384010	0.0266906
$208$	0	0
$209$	4.42426	0.306032
$210$	0	0
$211$	7.06980	0.486705	0.243353	−	0.969938i	$-0.421753\pi$
$211$	7.06980	0.486705	0.243353	+	0.969938i	$0.421753\pi$
$212$	0	0
$213$	27.4922	1.88373
$214$	0	0
$215$	0	0
$216$	0	0
$217$	33.0725	2.24511
$218$	0	0
$219$	−15.3872	−1.03977
$220$	0	0
$221$	6.94055	0.466872
$222$	0	0
$223$	−8.58817	−0.575106	−0.287553	−	0.957765i	$-0.592842\pi$
$223$	−8.58817	−0.575106	−0.287553	+	0.957765i	$0.592842\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.2068	−0.942937	−0.471469	−	0.881883i	$-0.656276\pi$
$227$	−14.2068	−0.942937	−0.471469	+	0.881883i	$0.656276\pi$
$228$	0	0
$229$	−7.49721	−0.495429	−0.247715	−	0.968833i	$-0.579680\pi$
$229$	−7.49721	−0.495429	−0.247715	+	0.968833i	$0.579680\pi$
$230$	0	0
$231$	15.3718	1.01139
$232$	0	0
$233$	−16.9301	−1.10912	−0.554562	−	0.832142i	$-0.687114\pi$
$233$	−16.9301	−1.10912	−0.554562	+	0.832142i	$0.687114\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−17.3131	−1.12460
$238$	0	0
$239$	17.7850	1.15042	0.575208	−	0.818007i	$-0.304921\pi$
$239$	17.7850	1.15042	0.575208	+	0.818007i	$0.304921\pi$
$240$	0	0
$241$	−9.55263	−0.615339	−0.307669	−	0.951493i	$-0.599549\pi$
$241$	−9.55263	−0.615339	−0.307669	+	0.951493i	$0.599549\pi$
$242$	0	0
$243$	3.96721	0.254497
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−11.2673	−0.716922
$248$	0	0
$249$	2.78153	0.176272
$250$	0	0
$251$	13.2075	0.833651	0.416826	−	0.908986i	$-0.363143\pi$
$251$	13.2075	0.833651	0.416826	+	0.908986i	$0.363143\pi$
$252$	0	0
$253$	−2.10339	−0.132239
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.41506	−0.213025	−0.106513	−	0.994311i	$-0.533968\pi$
$257$	−3.41506	−0.213025	−0.106513	+	0.994311i	$0.533968\pi$
$258$	0	0
$259$	20.2661	1.25927
$260$	0	0
$261$	2.31608	0.143362
$262$	0	0
$263$	−18.6589	−1.15056	−0.575279	−	0.817957i	$-0.695106\pi$
$263$	−18.6589	−1.15056	−0.575279	+	0.817957i	$0.695106\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−19.4815	−1.19225
$268$	0	0
$269$	−24.2224	−1.47687	−0.738434	−	0.674326i	$-0.764434\pi$
$269$	−24.2224	−1.47687	−0.738434	+	0.674326i	$0.764434\pi$
$270$	0	0
$271$	10.1201	0.614749	0.307375	−	0.951589i	$-0.400550\pi$
$271$	10.1201	0.614749	0.307375	+	0.951589i	$0.400550\pi$
$272$	0	0
$273$	−39.1475	−2.36932
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−23.5298	−1.41377	−0.706884	−	0.707329i	$-0.749900\pi$
$277$	−23.5298	−1.41377	−0.706884	+	0.707329i	$0.749900\pi$
$278$	0	0
$279$	3.19684	0.191390
$280$	0	0
$281$	1.77314	0.105776	0.0528882	−	0.998600i	$-0.483157\pi$
$281$	1.77314	0.105776	0.0528882	+	0.998600i	$0.483157\pi$
$282$	0	0
$283$	−5.96641	−0.354666	−0.177333	−	0.984151i	$-0.556747\pi$
$283$	−5.96641	−0.354666	−0.177333	+	0.984151i	$0.556747\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−33.1323	−1.95574
$288$	0	0
$289$	−15.3212	−0.901250
$290$	0	0
$291$	0.620765	0.0363899
$292$	0	0
$293$	12.2404	0.715090	0.357545	−	0.933896i	$-0.383614\pi$
$293$	12.2404	0.715090	0.357545	+	0.933896i	$0.383614\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−10.1221	−0.587346
$298$	0	0
$299$	5.35673	0.309788
$300$	0	0
$301$	−30.9178	−1.78207
$302$	0	0
$303$	11.3891	0.654288
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.82713	0.503791	0.251896	−	0.967754i	$-0.418946\pi$
$307$	8.82713	0.503791	0.251896	+	0.967754i	$0.418946\pi$
$308$	0	0
$309$	23.2276	1.32137
$310$	0	0
$311$	13.9202	0.789345	0.394672	−	0.918822i	$-0.370858\pi$
$311$	13.9202	0.789345	0.394672	+	0.918822i	$0.370858\pi$
$312$	0	0
$313$	23.5030	1.32847	0.664235	−	0.747523i	$-0.268757\pi$
$313$	23.5030	1.32847	0.664235	+	0.747523i	$0.268757\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.3279	0.748570	0.374285	−	0.927314i	$-0.377888\pi$
$317$	13.3279	0.748570	0.374285	+	0.927314i	$0.377888\pi$
$318$	0	0
$319$	−12.6862	−0.710290
$320$	0	0
$321$	23.5354	1.31362
$322$	0	0
$323$	−2.72530	−0.151640
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−26.3362	−1.45639
$328$	0	0
$329$	−45.2749	−2.49609
$330$	0	0
$331$	0.270137	0.0148481	0.00742404	−	0.999972i	$-0.497637\pi$
$331$	0.270137	0.0148481	0.00742404	+	0.999972i	$0.497637\pi$
$332$	0	0
$333$	1.95896	0.107350
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−20.0205	−1.09058	−0.545292	−	0.838246i	$-0.683581\pi$
$337$	−20.0205	−1.09058	−0.545292	+	0.838246i	$0.683581\pi$
$338$	0	0
$339$	−8.45213	−0.459057
$340$	0	0
$341$	−17.5105	−0.948247
$342$	0	0
$343$	7.08149	0.382364
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.1156	0.811448	0.405724	−	0.913996i	$-0.367019\pi$
$347$	15.1156	0.811448	0.405724	+	0.913996i	$0.367019\pi$
$348$	0	0
$349$	−14.5673	−0.779771	−0.389886	−	0.920863i	$-0.627485\pi$
$349$	−14.5673	−0.779771	−0.389886	+	0.920863i	$0.627485\pi$
$350$	0	0
$351$	25.7782	1.37594
$352$	0	0
$353$	18.5376	0.986656	0.493328	−	0.869843i	$-0.335780\pi$
$353$	18.5376	0.986656	0.493328	+	0.869843i	$0.335780\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−9.46886	−0.501145
$358$	0	0
$359$	24.1834	1.27635	0.638175	−	0.769891i	$-0.279689\pi$
$359$	24.1834	1.27635	0.638175	+	0.769891i	$0.279689\pi$
$360$	0	0
$361$	−14.5757	−0.767144
$362$	0	0
$363$	12.0965	0.634903
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.79604	−0.302551	−0.151275	−	0.988492i	$-0.548338\pi$
$367$	−5.79604	−0.302551	−0.151275	+	0.988492i	$0.548338\pi$
$368$	0	0
$369$	−3.20262	−0.166722
$370$	0	0
$371$	−2.27901	−0.118320
$372$	0	0
$373$	−25.5753	−1.32424	−0.662119	−	0.749399i	$-0.730343\pi$
$373$	−25.5753	−1.32424	−0.662119	+	0.749399i	$0.730343\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.3081	1.66395
$378$	0	0
$379$	9.82223	0.504534	0.252267	−	0.967658i	$-0.418824\pi$
$379$	9.82223	0.504534	0.252267	+	0.967658i	$0.418824\pi$
$380$	0	0
$381$	23.2932	1.19335
$382$	0	0
$383$	33.9797	1.73628	0.868141	−	0.496319i	$-0.165315\pi$
$383$	33.9797	1.73628	0.868141	+	0.496319i	$0.165315\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.98857	−0.151918
$388$	0	0
$389$	−24.6744	−1.25104	−0.625521	−	0.780207i	$-0.715114\pi$
$389$	−24.6744	−1.25104	−0.625521	+	0.780207i	$0.715114\pi$
$390$	0	0
$391$	1.29567	0.0655247
$392$	0	0
$393$	8.61599	0.434619
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.17340	−0.159268	−0.0796342	−	0.996824i	$-0.525375\pi$
$397$	−3.17340	−0.159268	−0.0796342	+	0.996824i	$0.525375\pi$
$398$	0	0
$399$	15.3718	0.769552
$400$	0	0
$401$	4.07720	0.203606	0.101803	−	0.994805i	$-0.467539\pi$
$401$	4.07720	0.203606	0.101803	+	0.994805i	$0.467539\pi$
$402$	0	0
$403$	44.5942	2.22140
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.7301	−0.531869
$408$	0	0
$409$	18.5426	0.916871	0.458435	−	0.888728i	$-0.348410\pi$
$409$	18.5426	0.916871	0.458435	+	0.888728i	$0.348410\pi$
$410$	0	0
$411$	0.861138	0.0424768
$412$	0	0
$413$	36.4676	1.79445
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.35982	0.0665906
$418$	0	0
$419$	3.22587	0.157594	0.0787970	−	0.996891i	$-0.474892\pi$
$419$	3.22587	0.157594	0.0787970	+	0.996891i	$0.474892\pi$
$420$	0	0
$421$	18.0595	0.880167	0.440084	−	0.897957i	$-0.354949\pi$
$421$	18.0595	0.880167	0.440084	+	0.897957i	$0.354949\pi$
$422$	0	0
$423$	−4.37635	−0.212785
$424$	0	0
$425$	0	0
$426$	0	0
$427$	53.6909	2.59828
$428$	0	0
$429$	20.7270	1.00071
$430$	0	0
$431$	9.62458	0.463600	0.231800	−	0.972763i	$-0.425539\pi$
$431$	9.62458	0.463600	0.231800	+	0.972763i	$0.425539\pi$
$432$	0	0
$433$	−1.83990	−0.0884200	−0.0442100	−	0.999022i	$-0.514077\pi$
$433$	−1.83990	−0.0884200	−0.0442100	+	0.999022i	$0.514077\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.10339	−0.100619
$438$	0	0
$439$	−6.73907	−0.321638	−0.160819	−	0.986984i	$-0.551414\pi$
$439$	−6.73907	−0.321638	−0.160819	+	0.986984i	$0.551414\pi$
$440$	0	0
$441$	3.37258	0.160599
$442$	0	0
$443$	−11.4099	−0.542103	−0.271051	−	0.962565i	$-0.587371\pi$
$443$	−11.4099	−0.542103	−0.271051	+	0.962565i	$0.587371\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−26.8398	−1.26948
$448$	0	0
$449$	27.8081	1.31235	0.656173	−	0.754610i	$-0.272174\pi$
$449$	27.8081	1.31235	0.656173	+	0.754610i	$0.272174\pi$
$450$	0	0
$451$	17.5422	0.826028
$452$	0	0
$453$	39.8974	1.87454
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.6835	1.24820	0.624102	−	0.781343i	$-0.285465\pi$
$457$	26.6835	1.24820	0.624102	+	0.781343i	$0.285465\pi$
$458$	0	0
$459$	6.23513	0.291031
$460$	0	0
$461$	−23.0398	−1.07307	−0.536534	−	0.843879i	$-0.680267\pi$
$461$	−23.0398	−1.07307	−0.536534	+	0.843879i	$0.680267\pi$
$462$	0	0
$463$	23.3998	1.08748	0.543740	−	0.839253i	$-0.317008\pi$
$463$	23.3998	1.08748	0.543740	+	0.839253i	$0.317008\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.29380	−0.198693	−0.0993467	−	0.995053i	$-0.531675\pi$
$467$	−4.29380	−0.198693	−0.0993467	+	0.995053i	$0.531675\pi$
$468$	0	0
$469$	62.8936	2.90416
$470$	0	0
$471$	−14.1877	−0.653735
$472$	0	0
$473$	16.3697	0.752680
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.220293	−0.0100865
$478$	0	0
$479$	26.6694	1.21856	0.609278	−	0.792957i	$-0.291459\pi$
$479$	26.6694	1.21856	0.609278	+	0.792957i	$0.291459\pi$
$480$	0	0
$481$	27.3264	1.24597
$482$	0	0
$483$	−7.30809	−0.332530
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.67114	0.392927	0.196463	−	0.980511i	$-0.437054\pi$
$487$	8.67114	0.392927	0.196463	+	0.980511i	$0.437054\pi$
$488$	0	0
$489$	−8.63083	−0.390300
$490$	0	0
$491$	25.0900	1.13229	0.566147	−	0.824304i	$-0.308434\pi$
$491$	25.0900	1.13229	0.566147	+	0.824304i	$0.308434\pi$
$492$	0	0
$493$	7.81456	0.351950
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−59.3720	−2.66320
$498$	0	0
$499$	34.9323	1.56379	0.781893	−	0.623412i	$-0.214254\pi$
$499$	34.9323	1.56379	0.781893	+	0.623412i	$0.214254\pi$
$500$	0	0
$501$	21.4747	0.959418
$502$	0	0
$503$	−25.7782	−1.14939	−0.574695	−	0.818367i	$-0.694879\pi$
$503$	−25.7782	−1.14939	−0.574695	+	0.818367i	$0.694879\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−28.8713	−1.28222
$508$	0	0
$509$	4.21667	0.186900	0.0934502	−	0.995624i	$-0.470210\pi$
$509$	4.21667	0.186900	0.0934502	+	0.995624i	$0.470210\pi$
$510$	0	0
$511$	33.2302	1.47002
$512$	0	0
$513$	−10.1221	−0.446903
$514$	0	0
$515$	0	0
$516$	0	0
$517$	23.9712	1.05425
$518$	0	0
$519$	15.3001	0.671600
$520$	0	0
$521$	29.3112	1.28415	0.642073	−	0.766644i	$-0.278075\pi$
$521$	29.3112	1.28415	0.642073	+	0.766644i	$0.278075\pi$
$522$	0	0
$523$	31.1549	1.36231	0.681154	−	0.732140i	$-0.261478\pi$
$523$	31.1549	1.36231	0.681154	+	0.732140i	$0.261478\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.7863	0.469858
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.52502	0.152973
$532$	0	0
$533$	−44.6748	−1.93508
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7.49655	−0.323500
$538$	0	0
$539$	−18.4731	−0.795692
$540$	0	0
$541$	15.7393	0.676683	0.338342	−	0.941023i	$-0.390134\pi$
$541$	15.7393	0.676683	0.338342	+	0.941023i	$0.390134\pi$
$542$	0	0
$543$	−4.64212	−0.199213
$544$	0	0
$545$	0	0
$546$	0	0
$547$	39.2935	1.68007	0.840034	−	0.542534i	$-0.182535\pi$
$547$	39.2935	1.68007	0.840034	+	0.542534i	$0.182535\pi$
$548$	0	0
$549$	5.18985	0.221497
$550$	0	0
$551$	−12.6862	−0.540450
$552$	0	0
$553$	37.3892	1.58995
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.1005	1.61437	0.807186	−	0.590297i	$-0.200990\pi$
$557$	38.1005	1.61437	0.807186	+	0.590297i	$0.200990\pi$
$558$	0	0
$559$	−41.6889	−1.76325
$560$	0	0
$561$	5.01337	0.211665
$562$	0	0
$563$	−32.6868	−1.37758	−0.688792	−	0.724959i	$-0.741859\pi$
$563$	−32.6868	−1.37758	−0.688792	+	0.724959i	$0.741859\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−39.7454	−1.66915
$568$	0	0
$569$	21.0256	0.881438	0.440719	−	0.897645i	$-0.354724\pi$
$569$	21.0256	0.881438	0.440719	+	0.897645i	$0.354724\pi$
$570$	0	0
$571$	7.43876	0.311303	0.155651	−	0.987812i	$-0.450252\pi$
$571$	7.43876	0.311303	0.155651	+	0.987812i	$0.450252\pi$
$572$	0	0
$573$	−44.0357	−1.83962
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−30.4148	−1.26618	−0.633091	−	0.774077i	$-0.718214\pi$
$577$	−30.4148	−1.26618	−0.633091	+	0.774077i	$0.718214\pi$
$578$	0	0
$579$	−34.5348	−1.43522
$580$	0	0
$581$	−6.00698	−0.249212
$582$	0	0
$583$	1.20664	0.0499739
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.7544	1.39319	0.696596	−	0.717464i	$-0.254697\pi$
$587$	33.7544	1.39319	0.696596	+	0.717464i	$0.254697\pi$
$588$	0	0
$589$	−17.5105	−0.721508
$590$	0	0
$591$	−29.2587	−1.20354
$592$	0	0
$593$	27.3980	1.12510	0.562550	−	0.826763i	$-0.309820\pi$
$593$	27.3980	1.12510	0.562550	+	0.826763i	$0.309820\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	18.1666	0.743507
$598$	0	0
$599$	−37.2927	−1.52374	−0.761869	−	0.647731i	$-0.775718\pi$
$599$	−37.2927	−1.52374	−0.761869	+	0.647731i	$0.775718\pi$
$600$	0	0
$601$	−2.26039	−0.0922032	−0.0461016	−	0.998937i	$-0.514680\pi$
$601$	−2.26039	−0.0922032	−0.0461016	+	0.998937i	$0.514680\pi$
$602$	0	0
$603$	6.07941	0.247573
$604$	0	0
$605$	0	0
$606$	0	0
$607$	23.4782	0.952951	0.476476	−	0.879188i	$-0.341914\pi$
$607$	23.4782	0.952951	0.476476	+	0.879188i	$0.341914\pi$
$608$	0	0
$609$	−44.0773	−1.78610
$610$	0	0
$611$	−61.0477	−2.46972
$612$	0	0
$613$	−25.0418	−1.01143	−0.505714	−	0.862701i	$-0.668771\pi$
$613$	−25.0418	−1.01143	−0.505714	+	0.862701i	$0.668771\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.2705	−0.735541	−0.367771	−	0.929916i	$-0.619879\pi$
$617$	−18.2705	−0.735541	−0.367771	+	0.929916i	$0.619879\pi$
$618$	0	0
$619$	−7.17743	−0.288485	−0.144243	−	0.989542i	$-0.546075\pi$
$619$	−7.17743	−0.288485	−0.144243	+	0.989542i	$0.546075\pi$
$620$	0	0
$621$	4.81229	0.193111
$622$	0	0
$623$	42.0721	1.68558
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−8.13872	−0.325029
$628$	0	0
$629$	6.60960	0.263542
$630$	0	0
$631$	28.7621	1.14500	0.572500	−	0.819904i	$-0.305974\pi$
$631$	28.7621	1.14500	0.572500	+	0.819904i	$0.305974\pi$
$632$	0	0
$633$	−13.0054	−0.516917
$634$	0	0
$635$	0	0
$636$	0	0
$637$	47.0457	1.86402
$638$	0	0
$639$	−5.73900	−0.227031
$640$	0	0
$641$	−40.7383	−1.60907	−0.804533	−	0.593907i	$-0.797585\pi$
$641$	−40.7383	−1.60907	−0.804533	+	0.593907i	$0.797585\pi$
$642$	0	0
$643$	8.24043	0.324971	0.162485	−	0.986711i	$-0.448049\pi$
$643$	8.24043	0.324971	0.162485	+	0.986711i	$0.448049\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.1134	1.30182	0.650911	−	0.759154i	$-0.274387\pi$
$647$	33.1134	1.30182	0.650911	+	0.759154i	$0.274387\pi$
$648$	0	0
$649$	−19.3081	−0.757909
$650$	0	0
$651$	−60.8391	−2.38447
$652$	0	0
$653$	14.5452	0.569199	0.284599	−	0.958647i	$-0.408139\pi$
$653$	14.5452	0.569199	0.284599	+	0.958647i	$0.408139\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.21209	0.125316
$658$	0	0
$659$	24.4498	0.952427	0.476214	−	0.879330i	$-0.342009\pi$
$659$	24.4498	0.952427	0.476214	+	0.879330i	$0.342009\pi$
$660$	0	0
$661$	1.43400	0.0557763	0.0278882	−	0.999611i	$-0.491122\pi$
$661$	1.43400	0.0557763	0.0278882	+	0.999611i	$0.491122\pi$
$662$	0	0
$663$	−12.7676	−0.495853
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.03130	0.233533
$668$	0	0
$669$	15.7985	0.610806
$670$	0	0
$671$	−28.4271	−1.09742
$672$	0	0
$673$	−3.39873	−0.131011	−0.0655056	−	0.997852i	$-0.520866\pi$
$673$	−3.39873	−0.131011	−0.0655056	+	0.997852i	$0.520866\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15.7751	−0.606287	−0.303144	−	0.952945i	$-0.598036\pi$
$677$	−15.7751	−0.606287	−0.303144	+	0.952945i	$0.598036\pi$
$678$	0	0
$679$	−1.34060	−0.0514475
$680$	0	0
$681$	26.1343	1.00147
$682$	0	0
$683$	4.99979	0.191312	0.0956559	−	0.995414i	$-0.469505\pi$
$683$	4.99979	0.191312	0.0956559	+	0.995414i	$0.469505\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13.7916	0.526183
$688$	0	0
$689$	−3.07296	−0.117071
$690$	0	0
$691$	−40.6864	−1.54779	−0.773893	−	0.633317i	$-0.781693\pi$
$691$	−40.6864	−1.54779	−0.773893	+	0.633317i	$0.781693\pi$
$692$	0	0
$693$	−3.20886	−0.121895
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−10.8058	−0.409298
$698$	0	0
$699$	31.1440	1.17797
$700$	0	0
$701$	−1.30372	−0.0492407	−0.0246204	−	0.999697i	$-0.507838\pi$
$701$	−1.30372	−0.0492407	−0.0246204	+	0.999697i	$0.507838\pi$
$702$	0	0
$703$	−10.7301	−0.404692
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−24.5959	−0.925024
$708$	0	0
$709$	52.5228	1.97253	0.986267	−	0.165157i	$-0.0528131\pi$
$709$	52.5228	1.97253	0.986267	+	0.165157i	$0.0528131\pi$
$710$	0	0
$711$	3.61411	0.135540
$712$	0	0
$713$	8.32489	0.311770
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−32.7167	−1.22183
$718$	0	0
$719$	24.2329	0.903735	0.451868	−	0.892085i	$-0.350758\pi$
$719$	24.2329	0.903735	0.451868	+	0.892085i	$0.350758\pi$
$720$	0	0
$721$	−50.1622	−1.86814
$722$	0	0
$723$	17.5727	0.653536
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−37.0943	−1.37575	−0.687875	−	0.725829i	$-0.741456\pi$
$727$	−37.0943	−1.37575	−0.687875	+	0.725829i	$0.741456\pi$
$728$	0	0
$729$	22.7158	0.841324
$730$	0	0
$731$	−10.0836	−0.372954
$732$	0	0
$733$	46.3065	1.71037	0.855184	−	0.518325i	$-0.173444\pi$
$733$	46.3065	1.71037	0.855184	+	0.518325i	$0.173444\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−33.2996	−1.22660
$738$	0	0
$739$	32.3516	1.19007	0.595037	−	0.803698i	$-0.297137\pi$
$739$	32.3516	1.19007	0.595037	+	0.803698i	$0.297137\pi$
$740$	0	0
$741$	20.7270	0.761425
$742$	0	0
$743$	1.73343	0.0635935	0.0317967	−	0.999494i	$-0.489877\pi$
$743$	1.73343	0.0635935	0.0317967	+	0.999494i	$0.489877\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−0.580645	−0.0212447
$748$	0	0
$749$	−50.8268	−1.85717
$750$	0	0
$751$	−42.9983	−1.56903	−0.784515	−	0.620110i	$-0.787088\pi$
$751$	−42.9983	−1.56903	−0.784515	+	0.620110i	$0.787088\pi$
$752$	0	0
$753$	−24.2961	−0.885400
$754$	0	0
$755$	0	0
$756$	0	0
$757$	48.3219	1.75629	0.878144	−	0.478397i	$-0.158782\pi$
$757$	48.3219	1.75629	0.878144	+	0.478397i	$0.158782\pi$
$758$	0	0
$759$	3.86933	0.140448
$760$	0	0
$761$	19.2528	0.697915	0.348958	−	0.937139i	$-0.386536\pi$
$761$	19.2528	0.697915	0.348958	+	0.937139i	$0.386536\pi$
$762$	0	0
$763$	56.8754	2.05903
$764$	0	0
$765$	0	0
$766$	0	0
$767$	49.1722	1.77550
$768$	0	0
$769$	9.05816	0.326645	0.163323	−	0.986573i	$-0.447779\pi$
$769$	9.05816	0.326645	0.163323	+	0.986573i	$0.447779\pi$
$770$	0	0
$771$	6.28223	0.226249
$772$	0	0
$773$	29.1276	1.04765	0.523824	−	0.851827i	$-0.324505\pi$
$773$	29.1276	1.04765	0.523824	+	0.851827i	$0.324505\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−37.2809	−1.33744
$778$	0	0
$779$	17.5422	0.628513
$780$	0	0
$781$	31.4350	1.12483
$782$	0	0
$783$	29.0244	1.03725
$784$	0	0
$785$	0	0
$786$	0	0
$787$	23.8771	0.851128	0.425564	−	0.904928i	$-0.360076\pi$
$787$	23.8771	0.851128	0.425564	+	0.904928i	$0.360076\pi$
$788$	0	0
$789$	34.3243	1.22198
$790$	0	0
$791$	18.2532	0.649008
$792$	0	0
$793$	72.3956	2.57084
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19.8896	−0.704526	−0.352263	−	0.935901i	$-0.614588\pi$
$797$	−19.8896	−0.704526	−0.352263	+	0.935901i	$0.614588\pi$
$798$	0	0
$799$	−14.7660	−0.522383
$800$	0	0
$801$	4.06677	0.143692
$802$	0	0
$803$	−17.5940	−0.620879
$804$	0	0
$805$	0	0
$806$	0	0
$807$	44.5588	1.56854
$808$	0	0
$809$	26.3598	0.926760	0.463380	−	0.886160i	$-0.346637\pi$
$809$	26.3598	0.926760	0.463380	+	0.886160i	$0.346637\pi$
$810$	0	0
$811$	14.2482	0.500323	0.250161	−	0.968204i	$-0.419516\pi$
$811$	14.2482	0.500323	0.250161	+	0.968204i	$0.419516\pi$
$812$	0	0
$813$	−18.6165	−0.652910
$814$	0	0
$815$	0	0
$816$	0	0
$817$	16.3697	0.572703
$818$	0	0
$819$	8.17205	0.285555
$820$	0	0
$821$	5.56520	0.194227	0.0971134	−	0.995273i	$-0.469039\pi$
$821$	5.56520	0.194227	0.0971134	+	0.995273i	$0.469039\pi$
$822$	0	0
$823$	34.1312	1.18974	0.594869	−	0.803823i	$-0.297204\pi$
$823$	34.1312	1.18974	0.594869	+	0.803823i	$0.297204\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.5085	0.817472	0.408736	−	0.912653i	$-0.365970\pi$
$827$	23.5085	0.817472	0.408736	+	0.912653i	$0.365970\pi$
$828$	0	0
$829$	−31.7442	−1.10252	−0.551260	−	0.834333i	$-0.685853\pi$
$829$	−31.7442	−1.10252	−0.551260	+	0.834333i	$0.685853\pi$
$830$	0	0
$831$	43.2846	1.50153
$832$	0	0
$833$	11.3792	0.394267
$834$	0	0
$835$	0	0
$836$	0	0
$837$	40.0618	1.38474
$838$	0	0
$839$	−24.5409	−0.847247	−0.423623	−	0.905838i	$-0.639242\pi$
$839$	−24.5409	−0.847247	−0.423623	+	0.905838i	$0.639242\pi$
$840$	0	0
$841$	7.37661	0.254366
$842$	0	0
$843$	−3.26180	−0.112343
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−26.1236	−0.897618
$848$	0	0
$849$	10.9756	0.376682
$850$	0	0
$851$	5.10131	0.174871
$852$	0	0
$853$	−23.7499	−0.813183	−0.406591	−	0.913610i	$-0.633283\pi$
$853$	−23.7499	−0.813183	−0.406591	+	0.913610i	$0.633283\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−43.0643	−1.47105	−0.735524	−	0.677499i	$-0.763064\pi$
$857$	−43.0643	−1.47105	−0.735524	+	0.677499i	$0.763064\pi$
$858$	0	0
$859$	25.0591	0.855006	0.427503	−	0.904014i	$-0.359393\pi$
$859$	25.0591	0.855006	0.427503	+	0.904014i	$0.359393\pi$
$860$	0	0
$861$	60.9490	2.07714
$862$	0	0
$863$	−46.0967	−1.56915	−0.784575	−	0.620033i	$-0.787119\pi$
$863$	−46.0967	−1.56915	−0.784575	+	0.620033i	$0.787119\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	28.1845	0.957195
$868$	0	0
$869$	−19.7960	−0.671535
$870$	0	0
$871$	84.8044	2.87349
$872$	0	0
$873$	−0.129585	−0.00438578
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−35.1644	−1.18742	−0.593709	−	0.804680i	$-0.702337\pi$
$877$	−35.1644	−1.18742	−0.593709	+	0.804680i	$0.702337\pi$
$878$	0	0
$879$	−22.5170	−0.759480
$880$	0	0
$881$	−38.8350	−1.30838	−0.654192	−	0.756329i	$-0.726991\pi$
$881$	−38.8350	−1.30838	−0.654192	+	0.756329i	$0.726991\pi$
$882$	0	0
$883$	−10.5954	−0.356562	−0.178281	−	0.983980i	$-0.557054\pi$
$883$	−10.5954	−0.356562	−0.178281	+	0.983980i	$0.557054\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.8733	0.902316	0.451158	−	0.892444i	$-0.351011\pi$
$887$	26.8733	0.902316	0.451158	+	0.892444i	$0.351011\pi$
$888$	0	0
$889$	−50.3040	−1.68714
$890$	0	0
$891$	21.0435	0.704984
$892$	0	0
$893$	23.9712	0.802165
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−9.85407	−0.329018
$898$	0	0
$899$	50.2099	1.67459
$900$	0	0
$901$	−0.743278	−0.0247622
$902$	0	0
$903$	56.8754	1.89270
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−58.3210	−1.93652	−0.968259	−	0.249949i	$-0.919586\pi$
$907$	−58.3210	−1.93652	−0.968259	+	0.249949i	$0.919586\pi$
$908$	0	0
$909$	−2.37748	−0.0788561
$910$	0	0
$911$	−40.3287	−1.33615	−0.668074	−	0.744095i	$-0.732881\pi$
$911$	−40.3287	−1.33615	−0.668074	+	0.744095i	$0.732881\pi$
$912$	0	0
$913$	3.18045	0.105257
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−18.6071	−0.614459
$918$	0	0
$919$	37.6333	1.24141	0.620704	−	0.784045i	$-0.286847\pi$
$919$	37.6333	1.24141	0.620704	+	0.784045i	$0.286847\pi$
$920$	0	0
$921$	−16.2381	−0.535064
$922$	0	0
$923$	−80.0559	−2.63507
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4.84877	−0.159254
$928$	0	0
$929$	30.7190	1.00786	0.503929	−	0.863745i	$-0.331887\pi$
$929$	30.7190	1.00786	0.503929	+	0.863745i	$0.331887\pi$
$930$	0	0
$931$	−18.4731	−0.605431
$932$	0	0
$933$	−25.6072	−0.838343
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−40.5843	−1.32583	−0.662915	−	0.748694i	$-0.730681\pi$
$937$	−40.5843	−1.32583	−0.662915	+	0.748694i	$0.730681\pi$
$938$	0	0
$939$	−43.2355	−1.41094
$940$	0	0
$941$	36.5104	1.19020	0.595102	−	0.803650i	$-0.297112\pi$
$941$	36.5104	1.19020	0.595102	+	0.803650i	$0.297112\pi$
$942$	0	0
$943$	−8.33994	−0.271586
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34.9681	−1.13631	−0.568155	−	0.822921i	$-0.692343\pi$
$947$	−34.9681	−1.13631	−0.568155	+	0.822921i	$0.692343\pi$
$948$	0	0
$949$	44.8069	1.45449
$950$	0	0
$951$	−24.5176	−0.795038
$952$	0	0
$953$	35.7164	1.15697	0.578484	−	0.815694i	$-0.303645\pi$
$953$	35.7164	1.15697	0.578484	+	0.815694i	$0.303645\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	23.3371	0.754382
$958$	0	0
$959$	−1.85971	−0.0600532
$960$	0	0
$961$	38.3038	1.23561
$962$	0	0
$963$	−4.91301	−0.158320
$964$	0	0
$965$	0	0
$966$	0	0
$967$	43.0422	1.38414	0.692072	−	0.721829i	$-0.256698\pi$
$967$	43.0422	1.38414	0.692072	+	0.721829i	$0.256698\pi$
$968$	0	0
$969$	5.01337	0.161053
$970$	0	0
$971$	9.76196	0.313276	0.156638	−	0.987656i	$-0.449934\pi$
$971$	9.76196	0.313276	0.156638	+	0.987656i	$0.449934\pi$
$972$	0	0
$973$	−2.93666	−0.0941449
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.0469	0.769329	0.384665	−	0.923056i	$-0.374317\pi$
$977$	24.0469	0.769329	0.384665	+	0.923056i	$0.374317\pi$
$978$	0	0
$979$	−22.2754	−0.711926
$980$	0	0
$981$	5.49768	0.175527
$982$	0	0
$983$	19.2418	0.613717	0.306859	−	0.951755i	$-0.400722\pi$
$983$	19.2418	0.613717	0.306859	+	0.951755i	$0.400722\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	83.2862	2.65103
$988$	0	0
$989$	−7.78253	−0.247470
$990$	0	0
$991$	38.1944	1.21328	0.606642	−	0.794975i	$-0.292516\pi$
$991$	38.1944	1.21328	0.606642	+	0.794975i	$0.292516\pi$
$992$	0	0
$993$	−0.496936	−0.0157698
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−41.5981	−1.31742	−0.658712	−	0.752395i	$-0.728899\pi$
$997$	−41.5981	−1.31742	−0.658712	+	0.752395i	$0.728899\pi$
$998$	0	0
$999$	24.5490	0.776696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.ct.1.2		5
4.3	odd	2		4600.2.a.bf.1.4	yes	5
5.4	even	2		9200.2.a.cv.1.4		5
20.3	even	4		4600.2.e.w.4049.8		10
20.7	even	4		4600.2.e.w.4049.3		10
20.19	odd	2		4600.2.a.bd.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bd.1.2	✓	5	20.19	odd	2
4600.2.a.bf.1.4	yes	5	4.3	odd	2
4600.2.e.w.4049.3		10	20.7	even	4
4600.2.e.w.4049.8		10	20.3	even	4
9200.2.a.ct.1.2		5	1.1	even	1	trivial
9200.2.a.cv.1.4		5	5.4	even	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q-1.83957 q^{3} +3.97272 q^{7} +0.384010 q^{9} +O(q^{10})\)	\(q-1.83957 q^{3} +3.97272 q^{7} +0.384010 q^{9} -2.10339 q^{11} +5.35673 q^{13} +1.29567 q^{17} -2.10339 q^{19} -7.30809 q^{21} +1.00000 q^{23} +4.81229 q^{27} +6.03130 q^{29} +8.32489 q^{31} +3.86933 q^{33} +5.10131 q^{37} -9.85407 q^{39} -8.33994 q^{41} -7.78253 q^{43} -11.3964 q^{47} +8.78253 q^{49} -2.38347 q^{51} -0.573664 q^{53} +3.86933 q^{57} +9.17951 q^{59} +13.5149 q^{61} +1.52557 q^{63} +15.8314 q^{67} -1.83957 q^{69} -14.9449 q^{71} +8.36459 q^{73} -8.35619 q^{77} +9.41149 q^{79} -10.0046 q^{81} -1.51206 q^{83} -11.0950 q^{87} +10.5903 q^{89} +21.2808 q^{91} -15.3142 q^{93} -0.337451 q^{97} -0.807724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 5 * q - 4 * q^7 + 3 * q^9	\( 5 q - 4 q^{7} + 3 q^{9} + 4 q^{13} + 6 q^{17} + 5 q^{23} - 9 q^{27} + 12 q^{29} + 18 q^{31} + 6 q^{33} + 10 q^{37} - 9 q^{39} - 6 q^{41} - 10 q^{43} - 22 q^{47} + 15 q^{49} + 6 q^{51} + 10 q^{53} + 6 q^{57} + q^{59} + 10 q^{61} - 8 q^{67} - 8 q^{71} + 6 q^{73} - 27 q^{81} + 2 q^{83} - 39 q^{87} + 14 q^{89} + 46 q^{91} - 3 q^{93} + 6 q^{97} + 6 q^{99}+O(q^{100}) \) 5 * q - 4 * q^7 + 3 * q^9 + 4 * q^13 + 6 * q^17 + 5 * q^23 - 9 * q^27 + 12 * q^29 + 18 * q^31 + 6 * q^33 + 10 * q^37 - 9 * q^39 - 6 * q^41 - 10 * q^43 - 22 * q^47 + 15 * q^49 + 6 * q^51 + 10 * q^53 + 6 * q^57 + q^59 + 10 * q^61 - 8 * q^67 - 8 * q^71 + 6 * q^73 - 27 * q^81 + 2 * q^83 - 39 * q^87 + 14 * q^89 + 46 * q^91 - 3 * q^93 + 6 * q^97 + 6 * q^99

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