LMFDB - Embedded newform 8624.2.a.ck.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	yes
Analytic conductor:	$68.8629867032$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.19869$ 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)

$f(q)$	$=$	$q-2.19869 q^{3} +0.635552 q^{5} +1.83424 q^{9} +O(q^{10})$	$q-2.19869 q^{3} +0.635552 q^{5} +1.83424 q^{9} +1.00000 q^{11} +1.80131 q^{13} -1.39738 q^{15} +2.83424 q^{17} -5.56314 q^{19} -2.16576 q^{23} -4.59607 q^{25} +2.56314 q^{27} -10.4303 q^{29} +6.43032 q^{31} -2.19869 q^{33} +6.06587 q^{37} -3.96052 q^{39} -7.53566 q^{41} +4.86718 q^{43} +1.16576 q^{45} -2.83424 q^{47} -6.23163 q^{51} +7.46980 q^{53} +0.635552 q^{55} +12.2316 q^{57} +11.8068 q^{59} +4.33151 q^{61} +1.14483 q^{65} +1.60262 q^{67} +4.76183 q^{69} -4.29204 q^{71} +15.9935 q^{73} +10.1053 q^{75} -4.76183 q^{79} -11.1383 q^{81} -9.23163 q^{83} +1.80131 q^{85} +22.9330 q^{87} -0.364448 q^{89} -14.1383 q^{93} -3.53566 q^{95} +2.59607 q^{97} +1.83424 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 2 q^{5}+O(q^{10}) $ 3 * q - q^3 + 2 * q^5	$ 3 q - q^{3} + 2 q^{5} + 3 q^{11} + 11 q^{13} + 7 q^{15} + 3 q^{17} - 11 q^{19} - 12 q^{23} + 3 q^{25} + 2 q^{27} - 9 q^{29} - 3 q^{31} - q^{33} - 4 q^{37} + 5 q^{39} + 5 q^{41} - 2 q^{43} + 9 q^{45} - 3 q^{47} - 2 q^{51} + 17 q^{53} + 2 q^{55} + 20 q^{57} + 8 q^{59} + 24 q^{61} + 15 q^{65} + 16 q^{67} + 3 q^{69} - 7 q^{71} + 20 q^{73} + 25 q^{75} - 3 q^{79} - 17 q^{81} - 11 q^{83} + 11 q^{85} + 30 q^{87} - q^{89} - 26 q^{93} + 17 q^{95} - 9 q^{97}+O(q^{100}) $ 3 * q - q^3 + 2 * q^5 + 3 * q^11 + 11 * q^13 + 7 * q^15 + 3 * q^17 - 11 * q^19 - 12 * q^23 + 3 * q^25 + 2 * q^27 - 9 * q^29 - 3 * q^31 - q^33 - 4 * q^37 + 5 * q^39 + 5 * q^41 - 2 * q^43 + 9 * q^45 - 3 * q^47 - 2 * q^51 + 17 * q^53 + 2 * q^55 + 20 * q^57 + 8 * q^59 + 24 * q^61 + 15 * q^65 + 16 * q^67 + 3 * q^69 - 7 * q^71 + 20 * q^73 + 25 * q^75 - 3 * q^79 - 17 * q^81 - 11 * q^83 + 11 * q^85 + 30 * q^87 - q^89 - 26 * q^93 + 17 * q^95 - 9 * q^97

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
$3$	−2.19869	−1.26941	−0.634707	−	0.772752i	$-0.718879\pi$
$3$	−2.19869	−1.26941	−0.634707	+	0.772752i	$0.718879\pi$
$4$	0	0
$5$	0.635552	0.284227	0.142114	−	0.989850i	$-0.454610\pi$
$5$	0.635552	0.284227	0.142114	+	0.989850i	$0.454610\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.83424	0.611414
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.80131	0.499593	0.249797	−	0.968298i	$-0.419636\pi$
$13$	1.80131	0.499593	0.249797	+	0.968298i	$0.419636\pi$
$14$	0	0
$15$	−1.39738	−0.360803
$16$	0	0
$17$	2.83424	0.687405	0.343702	−	0.939079i	$-0.388319\pi$
$17$	2.83424	0.687405	0.343702	+	0.939079i	$0.388319\pi$
$18$	0	0
$19$	−5.56314	−1.27627	−0.638136	−	0.769924i	$-0.720294\pi$
$19$	−5.56314	−1.27627	−0.638136	+	0.769924i	$0.720294\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.16576	−0.451592	−0.225796	−	0.974175i	$-0.572498\pi$
$23$	−2.16576	−0.451592	−0.225796	+	0.974175i	$0.572498\pi$
$24$	0	0
$25$	−4.59607	−0.919215
$26$	0	0
$27$	2.56314	0.493276
$28$	0	0
$29$	−10.4303	−1.93686	−0.968431	−	0.249283i	$-0.919805\pi$
$29$	−10.4303	−1.93686	−0.968431	+	0.249283i	$0.919805\pi$
$30$	0	0
$31$	6.43032	1.15492	0.577460	−	0.816419i	$-0.304044\pi$
$31$	6.43032	1.15492	0.577460	+	0.816419i	$0.304044\pi$
$32$	0	0
$33$	−2.19869	−0.382743
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.06587	0.997223	0.498611	−	0.866826i	$-0.333843\pi$
$37$	6.06587	0.997223	0.498611	+	0.866826i	$0.333843\pi$
$38$	0	0
$39$	−3.96052	−0.634191
$40$	0	0
$41$	−7.53566	−1.17687	−0.588436	−	0.808543i	$-0.700256\pi$
$41$	−7.53566	−1.17687	−0.588436	+	0.808543i	$0.700256\pi$
$42$	0	0
$43$	4.86718	0.742238	0.371119	−	0.928585i	$-0.378974\pi$
$43$	4.86718	0.742238	0.371119	+	0.928585i	$0.378974\pi$
$44$	0	0
$45$	1.16576	0.173781
$46$	0	0
$47$	−2.83424	−0.413417	−0.206708	−	0.978403i	$-0.566275\pi$
$47$	−2.83424	−0.413417	−0.206708	+	0.978403i	$0.566275\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−6.23163	−0.872602
$52$	0	0
$53$	7.46980	1.02606	0.513028	−	0.858372i	$-0.328524\pi$
$53$	7.46980	1.02606	0.513028	+	0.858372i	$0.328524\pi$
$54$	0	0
$55$	0.635552	0.0856978
$56$	0	0
$57$	12.2316	1.62012
$58$	0	0
$59$	11.8068	1.53711	0.768555	−	0.639784i	$-0.220976\pi$
$59$	11.8068	1.53711	0.768555	+	0.639784i	$0.220976\pi$
$60$	0	0
$61$	4.33151	0.554593	0.277297	−	0.960784i	$-0.410561\pi$
$61$	4.33151	0.554593	0.277297	+	0.960784i	$0.410561\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.14483	0.141998
$66$	0	0
$67$	1.60262	0.195791	0.0978954	−	0.995197i	$-0.468789\pi$
$67$	1.60262	0.195791	0.0978954	+	0.995197i	$0.468789\pi$
$68$	0	0
$69$	4.76183	0.573257
$70$	0	0
$71$	−4.29204	−0.509371	−0.254685	−	0.967024i	$-0.581972\pi$
$71$	−4.29204	−0.509371	−0.254685	+	0.967024i	$0.581972\pi$
$72$	0	0
$73$	15.9935	1.87189	0.935946	−	0.352143i	$-0.114547\pi$
$73$	15.9935	1.87189	0.935946	+	0.352143i	$0.114547\pi$
$74$	0	0
$75$	10.1053	1.16686
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.76183	−0.535748	−0.267874	−	0.963454i	$-0.586321\pi$
$79$	−4.76183	−0.535748	−0.267874	+	0.963454i	$0.586321\pi$
$80$	0	0
$81$	−11.1383	−1.23759
$82$	0	0
$83$	−9.23163	−1.01330	−0.506651	−	0.862151i	$-0.669117\pi$
$83$	−9.23163	−1.01330	−0.506651	+	0.862151i	$0.669117\pi$
$84$	0	0
$85$	1.80131	0.195379
$86$	0	0
$87$	22.9330	2.45868
$88$	0	0
$89$	−0.364448	−0.0386314	−0.0193157	−	0.999813i	$-0.506149\pi$
$89$	−0.364448	−0.0386314	−0.0193157	+	0.999813i	$0.506149\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−14.1383	−1.46607
$94$	0	0
$95$	−3.53566	−0.362751
$96$	0	0
$97$	2.59607	0.263591	0.131796	−	0.991277i	$-0.457926\pi$
$97$	2.59607	0.263591	0.131796	+	0.991277i	$0.457926\pi$
$98$	0	0
$99$	1.83424	0.184348
$100$	0	0
$101$	9.90011	0.985098	0.492549	−	0.870285i	$-0.336065\pi$
$101$	9.90011	0.985098	0.492549	+	0.870285i	$0.336065\pi$
$102$	0	0
$103$	6.23163	0.614020	0.307010	−	0.951706i	$-0.400671\pi$
$103$	6.23163	0.614020	0.307010	+	0.951706i	$0.400671\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.0988	−1.07296	−0.536481	−	0.843912i	$-0.680247\pi$
$107$	−11.0988	−1.07296	−0.536481	+	0.843912i	$0.680247\pi$
$108$	0	0
$109$	−14.3040	−1.37008	−0.685039	−	0.728506i	$-0.740215\pi$
$109$	−14.3040	−1.37008	−0.685039	+	0.728506i	$0.740215\pi$
$110$	0	0
$111$	−13.3370	−1.26589
$112$	0	0
$113$	−8.68942	−0.817432	−0.408716	−	0.912662i	$-0.634023\pi$
$113$	−8.68942	−0.817432	−0.408716	+	0.912662i	$0.634023\pi$
$114$	0	0
$115$	−1.37645	−0.128355
$116$	0	0
$117$	3.30404	0.305458
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	16.5686	1.49394
$124$	0	0
$125$	−6.09880	−0.545494
$126$	0	0
$127$	−20.9989	−1.86335	−0.931676	−	0.363290i	$-0.881654\pi$
$127$	−20.9989	−1.86335	−0.931676	+	0.363290i	$0.881654\pi$
$128$	0	0
$129$	−10.7014	−0.942208
$130$	0	0
$131$	−13.7014	−1.19710	−0.598549	−	0.801086i	$-0.704256\pi$
$131$	−13.7014	−1.19710	−0.598549	+	0.801086i	$0.704256\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.62901	0.140203
$136$	0	0
$137$	7.26456	0.620653	0.310327	−	0.950630i	$-0.399562\pi$
$137$	7.26456	0.620653	0.310327	+	0.950630i	$0.399562\pi$
$138$	0	0
$139$	−2.60262	−0.220751	−0.110376	−	0.993890i	$-0.535205\pi$
$139$	−2.60262	−0.220751	−0.110376	+	0.993890i	$0.535205\pi$
$140$	0	0
$141$	6.23163	0.524798
$142$	0	0
$143$	1.80131	0.150633
$144$	0	0
$145$	−6.62901	−0.550509
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.00000	−0.0819232	−0.0409616	−	0.999161i	$-0.513042\pi$
$149$	−1.00000	−0.0819232	−0.0409616	+	0.999161i	$0.513042\pi$
$150$	0	0
$151$	−1.73544	−0.141228	−0.0706140	−	0.997504i	$-0.522496\pi$
$151$	−1.73544	−0.141228	−0.0706140	+	0.997504i	$0.522496\pi$
$152$	0	0
$153$	5.19869	0.420289
$154$	0	0
$155$	4.08680	0.328260
$156$	0	0
$157$	7.93959	0.633648	0.316824	−	0.948484i	$-0.397383\pi$
$157$	7.93959	0.633648	0.316824	+	0.948484i	$0.397383\pi$
$158$	0	0
$159$	−16.4238	−1.30249
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.0055	1.25364	0.626822	−	0.779162i	$-0.284355\pi$
$163$	16.0055	1.25364	0.626822	+	0.779162i	$0.284355\pi$
$164$	0	0
$165$	−1.39738	−0.108786
$166$	0	0
$167$	1.15921	0.0897026	0.0448513	−	0.998994i	$-0.485719\pi$
$167$	1.15921	0.0897026	0.0448513	+	0.998994i	$0.485719\pi$
$168$	0	0
$169$	−9.75529	−0.750407
$170$	0	0
$171$	−10.2042	−0.780331
$172$	0	0
$173$	0.993456	0.0755311	0.0377655	−	0.999287i	$-0.487976\pi$
$173$	0.993456	0.0755311	0.0377655	+	0.999287i	$0.487976\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−25.9594	−1.95123
$178$	0	0
$179$	19.6619	1.46960	0.734801	−	0.678282i	$-0.237275\pi$
$179$	19.6619	1.46960	0.734801	+	0.678282i	$0.237275\pi$
$180$	0	0
$181$	23.8726	1.77444	0.887220	−	0.461347i	$-0.152634\pi$
$181$	23.8726	1.77444	0.887220	+	0.461347i	$0.152634\pi$
$182$	0	0
$183$	−9.52366	−0.704009
$184$	0	0
$185$	3.85517	0.283438
$186$	0	0
$187$	2.83424	0.207260
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.1317	0.805464	0.402732	−	0.915318i	$-0.368061\pi$
$191$	11.1317	0.805464	0.402732	+	0.915318i	$0.368061\pi$
$192$	0	0
$193$	3.61462	0.260186	0.130093	−	0.991502i	$-0.458472\pi$
$193$	3.61462	0.260186	0.130093	+	0.991502i	$0.458472\pi$
$194$	0	0
$195$	−2.51712	−0.180255
$196$	0	0
$197$	2.41831	0.172298	0.0861489	−	0.996282i	$-0.472544\pi$
$197$	2.41831	0.172298	0.0861489	+	0.996282i	$0.472544\pi$
$198$	0	0
$199$	−18.4962	−1.31116	−0.655580	−	0.755126i	$-0.727576\pi$
$199$	−18.4962	−1.31116	−0.655580	+	0.755126i	$0.727576\pi$
$200$	0	0
$201$	−3.52366	−0.248540
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4.78931	−0.334500
$206$	0	0
$207$	−3.97252	−0.276110
$208$	0	0
$209$	−5.56314	−0.384810
$210$	0	0
$211$	7.85517	0.540773	0.270386	−	0.962752i	$-0.412849\pi$
$211$	7.85517	0.540773	0.270386	+	0.962752i	$0.412849\pi$
$212$	0	0
$213$	9.43686	0.646603
$214$	0	0
$215$	3.09334	0.210964
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−35.1647	−2.37621
$220$	0	0
$221$	5.10535	0.343423
$222$	0	0
$223$	20.3370	1.36186	0.680932	−	0.732346i	$-0.261575\pi$
$223$	20.3370	1.36186	0.680932	+	0.732346i	$0.261575\pi$
$224$	0	0
$225$	−8.43032	−0.562021
$226$	0	0
$227$	7.21962	0.479183	0.239592	−	0.970874i	$-0.422986\pi$
$227$	7.21962	0.479183	0.239592	+	0.970874i	$0.422986\pi$
$228$	0	0
$229$	−12.0713	−0.797696	−0.398848	−	0.917017i	$-0.630590\pi$
$229$	−12.0713	−0.797696	−0.398848	+	0.917017i	$0.630590\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.54221	−0.494106	−0.247053	−	0.969002i	$-0.579462\pi$
$233$	−7.54221	−0.494106	−0.247053	+	0.969002i	$0.579462\pi$
$234$	0	0
$235$	−1.80131	−0.117504
$236$	0	0
$237$	10.4698	0.680086
$238$	0	0
$239$	9.84625	0.636901	0.318450	−	0.947940i	$-0.396838\pi$
$239$	9.84625	0.636901	0.318450	+	0.947940i	$0.396838\pi$
$240$	0	0
$241$	1.67503	0.107898	0.0539491	−	0.998544i	$-0.482819\pi$
$241$	1.67503	0.107898	0.0539491	+	0.998544i	$0.482819\pi$
$242$	0	0
$243$	16.8002	1.07773
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.0209	−0.637617
$248$	0	0
$249$	20.2975	1.28630
$250$	0	0
$251$	19.7738	1.24811	0.624057	−	0.781379i	$-0.285483\pi$
$251$	19.7738	1.24811	0.624057	+	0.781379i	$0.285483\pi$
$252$	0	0
$253$	−2.16576	−0.136160
$254$	0	0
$255$	−3.96052	−0.248017
$256$	0	0
$257$	−2.71997	−0.169667	−0.0848335	−	0.996395i	$-0.527036\pi$
$257$	−2.71997	−0.169667	−0.0848335	+	0.996395i	$0.527036\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−19.1317	−1.18422
$262$	0	0
$263$	−12.6949	−0.782800	−0.391400	−	0.920221i	$-0.628009\pi$
$263$	−12.6949	−0.782800	−0.391400	+	0.920221i	$0.628009\pi$
$264$	0	0
$265$	4.74744	0.291633
$266$	0	0
$267$	0.801309	0.0490393
$268$	0	0
$269$	7.09334	0.432489	0.216244	−	0.976339i	$-0.430619\pi$
$269$	7.09334	0.432489	0.216244	+	0.976339i	$0.430619\pi$
$270$	0	0
$271$	−18.2162	−1.10655	−0.553276	−	0.832998i	$-0.686623\pi$
$271$	−18.2162	−1.10655	−0.553276	+	0.832998i	$0.686623\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.59607	−0.277154
$276$	0	0
$277$	13.8486	0.832084	0.416042	−	0.909345i	$-0.363417\pi$
$277$	13.8486	0.832084	0.416042	+	0.909345i	$0.363417\pi$
$278$	0	0
$279$	11.7948	0.706134
$280$	0	0
$281$	21.0329	1.25472	0.627360	−	0.778730i	$-0.284136\pi$
$281$	21.0329	1.25472	0.627360	+	0.778730i	$0.284136\pi$
$282$	0	0
$283$	0.352445	0.0209507	0.0104753	−	0.999945i	$-0.496666\pi$
$283$	0.352445	0.0209507	0.0104753	+	0.999945i	$0.496666\pi$
$284$	0	0
$285$	7.77383	0.460482
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.96707	−0.527474
$290$	0	0
$291$	−5.70796	−0.334607
$292$	0	0
$293$	3.46325	0.202325	0.101163	−	0.994870i	$-0.467744\pi$
$293$	3.46325	0.202325	0.101163	+	0.994870i	$0.467744\pi$
$294$	0	0
$295$	7.50381	0.436889
$296$	0	0
$297$	2.56314	0.148728
$298$	0	0
$299$	−3.90120	−0.225612
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−21.7673	−1.25050
$304$	0	0
$305$	2.75290	0.157631
$306$	0	0
$307$	−6.51473	−0.371815	−0.185908	−	0.982567i	$-0.559523\pi$
$307$	−6.51473	−0.371815	−0.185908	+	0.982567i	$0.559523\pi$
$308$	0	0
$309$	−13.7014	−0.779447
$310$	0	0
$311$	11.6301	0.659482	0.329741	−	0.944071i	$-0.393039\pi$
$311$	11.6301	0.659482	0.329741	+	0.944071i	$0.393039\pi$
$312$	0	0
$313$	27.0977	1.53165	0.765827	−	0.643047i	$-0.222330\pi$
$313$	27.0977	1.53165	0.765827	+	0.643047i	$0.222330\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.86172	0.216896	0.108448	−	0.994102i	$-0.465412\pi$
$317$	3.86172	0.216896	0.108448	+	0.994102i	$0.465412\pi$
$318$	0	0
$319$	−10.4303	−0.583986
$320$	0	0
$321$	24.4028	1.36203
$322$	0	0
$323$	−15.7673	−0.877315
$324$	0	0
$325$	−8.27895	−0.459233
$326$	0	0
$327$	31.4502	1.73920
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−6.15028	−0.338050	−0.169025	−	0.985612i	$-0.554062\pi$
$331$	−6.15028	−0.338050	−0.169025	+	0.985612i	$0.554062\pi$
$332$	0	0
$333$	11.1263	0.609716
$334$	0	0
$335$	1.01855	0.0556491
$336$	0	0
$337$	−11.7607	−0.640649	−0.320324	−	0.947308i	$-0.603792\pi$
$337$	−11.7607	−0.640649	−0.320324	+	0.947308i	$0.603792\pi$
$338$	0	0
$339$	19.1053	1.03766
$340$	0	0
$341$	6.43032	0.348221
$342$	0	0
$343$	0	0
$344$	0	0
$345$	3.02639	0.162935
$346$	0	0
$347$	0.840787	0.0451358	0.0225679	−	0.999745i	$-0.492816\pi$
$347$	0.840787	0.0451358	0.0225679	+	0.999745i	$0.492816\pi$
$348$	0	0
$349$	−9.13174	−0.488811	−0.244405	−	0.969673i	$-0.578593\pi$
$349$	−9.13174	−0.488811	−0.244405	+	0.969673i	$0.578593\pi$
$350$	0	0
$351$	4.61701	0.246438
$352$	0	0
$353$	−22.7278	−1.20968	−0.604840	−	0.796347i	$-0.706763\pi$
$353$	−22.7278	−1.20968	−0.604840	+	0.796347i	$0.706763\pi$
$354$	0	0
$355$	−2.72781	−0.144777
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.2185	1.38376	0.691881	−	0.722012i	$-0.256782\pi$
$359$	26.2185	1.38376	0.691881	+	0.722012i	$0.256782\pi$
$360$	0	0
$361$	11.9485	0.628869
$362$	0	0
$363$	−2.19869	−0.115401
$364$	0	0
$365$	10.1647	0.532043
$366$	0	0
$367$	−3.82224	−0.199519	−0.0997597	−	0.995012i	$-0.531807\pi$
$367$	−3.82224	−0.199519	−0.0997597	+	0.995012i	$0.531807\pi$
$368$	0	0
$369$	−13.8222	−0.719557
$370$	0	0
$371$	0	0
$372$	0	0
$373$	15.1077	0.782249	0.391124	−	0.920338i	$-0.372086\pi$
$373$	15.1077	0.782249	0.391124	+	0.920338i	$0.372086\pi$
$374$	0	0
$375$	13.4094	0.692458
$376$	0	0
$377$	−18.7882	−0.967643
$378$	0	0
$379$	11.3765	0.584369	0.292185	−	0.956362i	$-0.405618\pi$
$379$	11.3765	0.584369	0.292185	+	0.956362i	$0.405618\pi$
$380$	0	0
$381$	46.1701	2.36537
$382$	0	0
$383$	8.88572	0.454039	0.227020	−	0.973890i	$-0.427102\pi$
$383$	8.88572	0.454039	0.227020	+	0.973890i	$0.427102\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.92759	0.453815
$388$	0	0
$389$	−19.8990	−1.00892	−0.504460	−	0.863435i	$-0.668309\pi$
$389$	−19.8990	−1.00892	−0.504460	+	0.863435i	$0.668309\pi$
$390$	0	0
$391$	−6.13828	−0.310426
$392$	0	0
$393$	30.1252	1.51962
$394$	0	0
$395$	−3.02639	−0.152274
$396$	0	0
$397$	34.8606	1.74961	0.874803	−	0.484480i	$-0.160991\pi$
$397$	34.8606	1.74961	0.874803	+	0.484480i	$0.160991\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.3963	0.569104	0.284552	−	0.958661i	$-0.408155\pi$
$401$	11.3963	0.569104	0.284552	+	0.958661i	$0.408155\pi$
$402$	0	0
$403$	11.5830	0.576990
$404$	0	0
$405$	−7.07896	−0.351756
$406$	0	0
$407$	6.06587	0.300674
$408$	0	0
$409$	4.08789	0.202133	0.101066	−	0.994880i	$-0.467775\pi$
$409$	4.08789	0.202133	0.101066	+	0.994880i	$0.467775\pi$
$410$	0	0
$411$	−15.9725	−0.787867
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5.86718	−0.288008
$416$	0	0
$417$	5.72235	0.280225
$418$	0	0
$419$	32.8002	1.60240	0.801198	−	0.598399i	$-0.204196\pi$
$419$	32.8002	1.60240	0.801198	+	0.598399i	$0.204196\pi$
$420$	0	0
$421$	−8.52128	−0.415302	−0.207651	−	0.978203i	$-0.566582\pi$
$421$	−8.52128	−0.415302	−0.207651	+	0.978203i	$0.566582\pi$
$422$	0	0
$423$	−5.19869	−0.252769
$424$	0	0
$425$	−13.0264	−0.631873
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−3.96052	−0.191216
$430$	0	0
$431$	16.7344	0.806066	0.403033	−	0.915186i	$-0.367956\pi$
$431$	16.7344	0.806066	0.403033	+	0.915186i	$0.367956\pi$
$432$	0	0
$433$	25.8661	1.24305	0.621523	−	0.783396i	$-0.286514\pi$
$433$	25.8661	1.24305	0.621523	+	0.783396i	$0.286514\pi$
$434$	0	0
$435$	14.5751	0.698825
$436$	0	0
$437$	12.0484	0.576353
$438$	0	0
$439$	−9.56860	−0.456684	−0.228342	−	0.973581i	$-0.573330\pi$
$439$	−9.56860	−0.456684	−0.228342	+	0.973581i	$0.573330\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−19.0329	−0.904282	−0.452141	−	0.891946i	$-0.649340\pi$
$443$	−19.0329	−0.904282	−0.452141	+	0.891946i	$0.649340\pi$
$444$	0	0
$445$	−0.231626	−0.0109801
$446$	0	0
$447$	2.19869	0.103995
$448$	0	0
$449$	33.3424	1.57353	0.786763	−	0.617255i	$-0.211755\pi$
$449$	33.3424	1.57353	0.786763	+	0.617255i	$0.211755\pi$
$450$	0	0
$451$	−7.53566	−0.354841
$452$	0	0
$453$	3.81570	0.179277
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.9605	0.559490	0.279745	−	0.960074i	$-0.409750\pi$
$457$	11.9605	0.559490	0.279745	+	0.960074i	$0.409750\pi$
$458$	0	0
$459$	7.26456	0.339081
$460$	0	0
$461$	−12.4896	−0.581701	−0.290850	−	0.956769i	$-0.593938\pi$
$461$	−12.4896	−0.581701	−0.290850	+	0.956769i	$0.593938\pi$
$462$	0	0
$463$	−12.3095	−0.572071	−0.286035	−	0.958219i	$-0.592338\pi$
$463$	−12.3095	−0.572071	−0.286035	+	0.958219i	$0.592338\pi$
$464$	0	0
$465$	−8.98561	−0.416698
$466$	0	0
$467$	32.7607	1.51599	0.757993	−	0.652262i	$-0.226180\pi$
$467$	32.7607	1.51599	0.757993	+	0.652262i	$0.226180\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−17.4567	−0.804363
$472$	0	0
$473$	4.86718	0.223793
$474$	0	0
$475$	25.5686	1.17317
$476$	0	0
$477$	13.7014	0.627345
$478$	0	0
$479$	25.9780	1.18696	0.593482	−	0.804847i	$-0.297753\pi$
$479$	25.9780	1.18696	0.593482	+	0.804847i	$0.297753\pi$
$480$	0	0
$481$	10.9265	0.498206
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.64994	0.0749199
$486$	0	0
$487$	−14.5357	−0.658674	−0.329337	−	0.944212i	$-0.606825\pi$
$487$	−14.5357	−0.658674	−0.329337	+	0.944212i	$0.606825\pi$
$488$	0	0
$489$	−35.1911	−1.59139
$490$	0	0
$491$	39.3952	1.77788	0.888941	−	0.458023i	$-0.151442\pi$
$491$	39.3952	1.77788	0.888941	+	0.458023i	$0.151442\pi$
$492$	0	0
$493$	−29.5621	−1.33141
$494$	0	0
$495$	1.16576	0.0523969
$496$	0	0
$497$	0	0
$498$	0	0
$499$	26.2305	1.17424	0.587120	−	0.809500i	$-0.300262\pi$
$499$	26.2305	1.17424	0.587120	+	0.809500i	$0.300262\pi$
$500$	0	0
$501$	−2.54875	−0.113870
$502$	0	0
$503$	3.94613	0.175949	0.0879747	−	0.996123i	$-0.471961\pi$
$503$	3.94613	0.175949	0.0879747	+	0.996123i	$0.471961\pi$
$504$	0	0
$505$	6.29204	0.279992
$506$	0	0
$507$	21.4489	0.952577
$508$	0	0
$509$	−16.9056	−0.749326	−0.374663	−	0.927161i	$-0.622242\pi$
$509$	−16.9056	−0.749326	−0.374663	+	0.927161i	$0.622242\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−14.2591	−0.629555
$514$	0	0
$515$	3.96052	0.174521
$516$	0	0
$517$	−2.83424	−0.124650
$518$	0	0
$519$	−2.18430	−0.0958803
$520$	0	0
$521$	−1.55768	−0.0682432	−0.0341216	−	0.999418i	$-0.510863\pi$
$521$	−1.55768	−0.0682432	−0.0341216	+	0.999418i	$0.510863\pi$
$522$	0	0
$523$	−12.5027	−0.546706	−0.273353	−	0.961914i	$-0.588133\pi$
$523$	−12.5027	−0.546706	−0.273353	+	0.961914i	$0.588133\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.2251	0.793897
$528$	0	0
$529$	−18.3095	−0.796065
$530$	0	0
$531$	21.6565	0.939811
$532$	0	0
$533$	−13.5741	−0.587958
$534$	0	0
$535$	−7.05387	−0.304965
$536$	0	0
$537$	−43.2305	−1.86554
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−16.5621	−0.712058	−0.356029	−	0.934475i	$-0.615870\pi$
$541$	−16.5621	−0.712058	−0.356029	+	0.934475i	$0.615870\pi$
$542$	0	0
$543$	−52.4886	−2.25250
$544$	0	0
$545$	−9.09096	−0.389414
$546$	0	0
$547$	6.64448	0.284097	0.142049	−	0.989860i	$-0.454631\pi$
$547$	6.64448	0.284097	0.142049	+	0.989860i	$0.454631\pi$
$548$	0	0
$549$	7.94505	0.339086
$550$	0	0
$551$	58.0253	2.47196
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−8.47634	−0.359801
$556$	0	0
$557$	13.1119	0.555569	0.277784	−	0.960643i	$-0.410400\pi$
$557$	13.1119	0.555569	0.277784	+	0.960643i	$0.410400\pi$
$558$	0	0
$559$	8.76729	0.370817
$560$	0	0
$561$	−6.23163	−0.263099
$562$	0	0
$563$	30.4886	1.28494	0.642470	−	0.766311i	$-0.277910\pi$
$563$	30.4886	1.28494	0.642470	+	0.766311i	$0.277910\pi$
$564$	0	0
$565$	−5.52258	−0.232337
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−35.3579	−1.48228	−0.741140	−	0.671350i	$-0.765715\pi$
$569$	−35.3579	−1.48228	−0.741140	+	0.671350i	$0.765715\pi$
$570$	0	0
$571$	41.2844	1.72770	0.863849	−	0.503750i	$-0.168047\pi$
$571$	41.2844	1.72770	0.863849	+	0.503750i	$0.168047\pi$
$572$	0	0
$573$	−24.4753	−1.02247
$574$	0	0
$575$	9.95398	0.415110
$576$	0	0
$577$	−8.70251	−0.362290	−0.181145	−	0.983456i	$-0.557980\pi$
$577$	−8.70251	−0.362290	−0.181145	+	0.983456i	$0.557980\pi$
$578$	0	0
$579$	−7.94743	−0.330284
$580$	0	0
$581$	0	0
$582$	0	0
$583$	7.46980	0.309367
$584$	0	0
$585$	2.09989	0.0868197
$586$	0	0
$587$	−23.0539	−0.951535	−0.475767	−	0.879571i	$-0.657830\pi$
$587$	−23.0539	−0.951535	−0.475767	+	0.879571i	$0.657830\pi$
$588$	0	0
$589$	−35.7727	−1.47399
$590$	0	0
$591$	−5.31713	−0.218717
$592$	0	0
$593$	−30.0988	−1.23601	−0.618005	−	0.786174i	$-0.712059\pi$
$593$	−30.0988	−1.23601	−0.618005	+	0.786174i	$0.712059\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	40.6674	1.66441
$598$	0	0
$599$	−29.2515	−1.19518	−0.597591	−	0.801801i	$-0.703875\pi$
$599$	−29.2515	−1.19518	−0.597591	+	0.801801i	$0.703875\pi$
$600$	0	0
$601$	26.8222	1.09410	0.547051	−	0.837099i	$-0.315750\pi$
$601$	26.8222	1.09410	0.547051	+	0.837099i	$0.315750\pi$
$602$	0	0
$603$	2.93959	0.119709
$604$	0	0
$605$	0.635552	0.0258389
$606$	0	0
$607$	32.2096	1.30735	0.653674	−	0.756776i	$-0.273227\pi$
$607$	32.2096	1.30735	0.653674	+	0.756776i	$0.273227\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.10535	−0.206540
$612$	0	0
$613$	44.5950	1.80117	0.900587	−	0.434675i	$-0.143137\pi$
$613$	44.5950	1.80117	0.900587	+	0.434675i	$0.143137\pi$
$614$	0	0
$615$	10.5302	0.424619
$616$	0	0
$617$	−0.531290	−0.0213889	−0.0106945	−	0.999943i	$-0.503404\pi$
$617$	−0.531290	−0.0213889	−0.0106945	+	0.999943i	$0.503404\pi$
$618$	0	0
$619$	41.7453	1.67788	0.838942	−	0.544221i	$-0.183175\pi$
$619$	41.7453	1.67788	0.838942	+	0.544221i	$0.183175\pi$
$620$	0	0
$621$	−5.55114	−0.222759
$622$	0	0
$623$	0	0
$624$	0	0
$625$	19.1043	0.764170
$626$	0	0
$627$	12.2316	0.488484
$628$	0	0
$629$	17.1921	0.685496
$630$	0	0
$631$	0.217238	0.00864810	0.00432405	−	0.999991i	$-0.498624\pi$
$631$	0.217238	0.00864810	0.00432405	+	0.999991i	$0.498624\pi$
$632$	0	0
$633$	−17.2711	−0.686465
$634$	0	0
$635$	−13.3459	−0.529616
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−7.87264	−0.311437
$640$	0	0
$641$	−9.69050	−0.382752	−0.191376	−	0.981517i	$-0.561295\pi$
$641$	−9.69050	−0.382752	−0.191376	+	0.981517i	$0.561295\pi$
$642$	0	0
$643$	16.4633	0.649247	0.324624	−	0.945843i	$-0.394762\pi$
$643$	16.4633	0.649247	0.324624	+	0.945843i	$0.394762\pi$
$644$	0	0
$645$	−6.80131	−0.267801
$646$	0	0
$647$	0.873721	0.0343495	0.0171748	−	0.999853i	$-0.494533\pi$
$647$	0.873721	0.0343495	0.0171748	+	0.999853i	$0.494533\pi$
$648$	0	0
$649$	11.8068	0.463456
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−39.9330	−1.56270	−0.781350	−	0.624093i	$-0.785469\pi$
$653$	−39.9330	−1.56270	−0.781350	+	0.624093i	$0.785469\pi$
$654$	0	0
$655$	−8.70796	−0.340248
$656$	0	0
$657$	29.3359	1.14450
$658$	0	0
$659$	6.89465	0.268578	0.134289	−	0.990942i	$-0.457125\pi$
$659$	6.89465	0.268578	0.134289	+	0.990942i	$0.457125\pi$
$660$	0	0
$661$	40.0144	1.55638	0.778190	−	0.628029i	$-0.216138\pi$
$661$	40.0144	1.55638	0.778190	+	0.628029i	$0.216138\pi$
$662$	0	0
$663$	−11.2251	−0.435946
$664$	0	0
$665$	0	0
$666$	0	0
$667$	22.5895	0.874670
$668$	0	0
$669$	−44.7147	−1.72877
$670$	0	0
$671$	4.33151	0.167216
$672$	0	0
$673$	31.3788	1.20957	0.604783	−	0.796391i	$-0.293260\pi$
$673$	31.3788	1.20957	0.604783	+	0.796391i	$0.293260\pi$
$674$	0	0
$675$	−11.7804	−0.453427
$676$	0	0
$677$	36.5315	1.40402	0.702010	−	0.712167i	$-0.252286\pi$
$677$	36.5315	1.40402	0.702010	+	0.712167i	$0.252286\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−15.8737	−0.608282
$682$	0	0
$683$	15.2700	0.584291	0.292146	−	0.956374i	$-0.405631\pi$
$683$	15.2700	0.584291	0.292146	+	0.956374i	$0.405631\pi$
$684$	0	0
$685$	4.61701	0.176407
$686$	0	0
$687$	26.5411	1.01261
$688$	0	0
$689$	13.4554	0.512610
$690$	0	0
$691$	−38.3843	−1.46021	−0.730104	−	0.683336i	$-0.760528\pi$
$691$	−38.3843	−1.46021	−0.730104	+	0.683336i	$0.760528\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.65410	−0.0627435
$696$	0	0
$697$	−21.3579	−0.808988
$698$	0	0
$699$	16.5830	0.627226
$700$	0	0
$701$	17.9056	0.676284	0.338142	−	0.941095i	$-0.390202\pi$
$701$	17.9056	0.676284	0.338142	+	0.941095i	$0.390202\pi$
$702$	0	0
$703$	−33.7453	−1.27273
$704$	0	0
$705$	3.96052	0.149162
$706$	0	0
$707$	0	0
$708$	0	0
$709$	34.3994	1.29190	0.645948	−	0.763382i	$-0.276462\pi$
$709$	34.3994	1.29190	0.645948	+	0.763382i	$0.276462\pi$
$710$	0	0
$711$	−8.73436	−0.327564
$712$	0	0
$713$	−13.9265	−0.521552
$714$	0	0
$715$	1.14483	0.0428140
$716$	0	0
$717$	−21.6489	−0.808491
$718$	0	0
$719$	49.4172	1.84295	0.921476	−	0.388436i	$-0.126984\pi$
$719$	49.4172	1.84295	0.921476	+	0.388436i	$0.126984\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−3.68287	−0.136968
$724$	0	0
$725$	47.9385	1.78039
$726$	0	0
$727$	−19.8201	−0.735086	−0.367543	−	0.930007i	$-0.619801\pi$
$727$	−19.8201	−0.735086	−0.367543	+	0.930007i	$0.619801\pi$
$728$	0	0
$729$	−3.52366	−0.130506
$730$	0	0
$731$	13.7948	0.510218
$732$	0	0
$733$	30.6476	1.13199	0.565997	−	0.824408i	$-0.308492\pi$
$733$	30.6476	1.13199	0.565997	+	0.824408i	$0.308492\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.60262	0.0590332
$738$	0	0
$739$	−17.1019	−0.629103	−0.314551	−	0.949240i	$-0.601854\pi$
$739$	−17.1019	−0.629103	−0.314551	+	0.949240i	$0.601854\pi$
$740$	0	0
$741$	22.0329	0.809400
$742$	0	0
$743$	6.97252	0.255797	0.127899	−	0.991787i	$-0.459177\pi$
$743$	6.97252	0.255797	0.127899	+	0.991787i	$0.459177\pi$
$744$	0	0
$745$	−0.635552	−0.0232848
$746$	0	0
$747$	−16.9330	−0.619548
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−15.2327	−0.555849	−0.277925	−	0.960603i	$-0.589647\pi$
$751$	−15.2327	−0.555849	−0.277925	+	0.960603i	$0.589647\pi$
$752$	0	0
$753$	−43.4766	−1.58437
$754$	0	0
$755$	−1.10296	−0.0401409
$756$	0	0
$757$	−14.5326	−0.528196	−0.264098	−	0.964496i	$-0.585074\pi$
$757$	−14.5326	−0.528196	−0.264098	+	0.964496i	$0.585074\pi$
$758$	0	0
$759$	4.76183	0.172843
$760$	0	0
$761$	1.71342	0.0621116	0.0310558	−	0.999518i	$-0.490113\pi$
$761$	1.71342	0.0621116	0.0310558	+	0.999518i	$0.490113\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3.30404	0.119458
$766$	0	0
$767$	21.2676	0.767930
$768$	0	0
$769$	36.5874	1.31937	0.659687	−	0.751540i	$-0.270689\pi$
$769$	36.5874	1.31937	0.659687	+	0.751540i	$0.270689\pi$
$770$	0	0
$771$	5.98037	0.215378
$772$	0	0
$773$	−26.8212	−0.964690	−0.482345	−	0.875981i	$-0.660215\pi$
$773$	−26.8212	−0.964690	−0.482345	+	0.875981i	$0.660215\pi$
$774$	0	0
$775$	−29.5542	−1.06162
$776$	0	0
$777$	0	0
$778$	0	0
$779$	41.9219	1.50201
$780$	0	0
$781$	−4.29204	−0.153581
$782$	0	0
$783$	−26.7344	−0.955408
$784$	0	0
$785$	5.04602	0.180100
$786$	0	0
$787$	4.95159	0.176505	0.0882526	−	0.996098i	$-0.471872\pi$
$787$	4.95159	0.176505	0.0882526	+	0.996098i	$0.471872\pi$
$788$	0	0
$789$	27.9121	0.993698
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7.80239	0.277071
$794$	0	0
$795$	−10.4382	−0.370203
$796$	0	0
$797$	26.6818	0.945117	0.472559	−	0.881299i	$-0.343330\pi$
$797$	26.6818	0.945117	0.472559	+	0.881299i	$0.343330\pi$
$798$	0	0
$799$	−8.03293	−0.284185
$800$	0	0
$801$	−0.668486	−0.0236198
$802$	0	0
$803$	15.9935	0.564397
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−15.5961	−0.549008
$808$	0	0
$809$	−39.5400	−1.39015	−0.695077	−	0.718935i	$-0.744630\pi$
$809$	−39.5400	−1.39015	−0.695077	+	0.718935i	$0.744630\pi$
$810$	0	0
$811$	−33.4543	−1.17474	−0.587370	−	0.809318i	$-0.699837\pi$
$811$	−33.4543	−1.17474	−0.587370	+	0.809318i	$0.699837\pi$
$812$	0	0
$813$	40.0517	1.40467
$814$	0	0
$815$	10.1723	0.356320
$816$	0	0
$817$	−27.0768	−0.947297
$818$	0	0
$819$	0	0
$820$	0	0
$821$	56.8619	1.98450	0.992248	−	0.124277i	$-0.0396611\pi$
$821$	56.8619	1.98450	0.992248	+	0.124277i	$0.0396611\pi$
$822$	0	0
$823$	40.1756	1.40043	0.700217	−	0.713931i	$-0.253087\pi$
$823$	40.1756	1.40043	0.700217	+	0.713931i	$0.253087\pi$
$824$	0	0
$825$	10.1053	0.351823
$826$	0	0
$827$	5.73544	0.199441	0.0997204	−	0.995015i	$-0.468205\pi$
$827$	5.73544	0.199441	0.0997204	+	0.995015i	$0.468205\pi$
$828$	0	0
$829$	35.5961	1.23630	0.618151	−	0.786059i	$-0.287882\pi$
$829$	35.5961	1.23630	0.618151	+	0.786059i	$0.287882\pi$
$830$	0	0
$831$	−30.4489	−1.05626
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0.736740	0.0254959
$836$	0	0
$837$	16.4818	0.569694
$838$	0	0
$839$	−41.7727	−1.44216	−0.721078	−	0.692854i	$-0.756353\pi$
$839$	−41.7727	−1.44216	−0.721078	+	0.692854i	$0.756353\pi$
$840$	0	0
$841$	79.7915	2.75143
$842$	0	0
$843$	−46.2449	−1.59276
$844$	0	0
$845$	−6.19999	−0.213286
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−0.774918	−0.0265951
$850$	0	0
$851$	−13.1372	−0.450337
$852$	0	0
$853$	−49.7871	−1.70468	−0.852340	−	0.522989i	$-0.824817\pi$
$853$	−49.7871	−1.70468	−0.852340	+	0.522989i	$0.824817\pi$
$854$	0	0
$855$	−6.48527	−0.221791
$856$	0	0
$857$	−25.4787	−0.870337	−0.435168	−	0.900349i	$-0.643311\pi$
$857$	−25.4787	−0.870337	−0.435168	+	0.900349i	$0.643311\pi$
$858$	0	0
$859$	16.1616	0.551427	0.275713	−	0.961240i	$-0.411086\pi$
$859$	16.1616	0.551427	0.275713	+	0.961240i	$0.411086\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.89903	−0.0986840	−0.0493420	−	0.998782i	$-0.515712\pi$
$863$	−2.89903	−0.0986840	−0.0493420	+	0.998782i	$0.515712\pi$
$864$	0	0
$865$	0.631393	0.0214680
$866$	0	0
$867$	19.7158	0.669584
$868$	0	0
$869$	−4.76183	−0.161534
$870$	0	0
$871$	2.88681	0.0978158
$872$	0	0
$873$	4.76183	0.161164
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−8.41484	−0.284149	−0.142075	−	0.989856i	$-0.545377\pi$
$877$	−8.41484	−0.284149	−0.142075	+	0.989856i	$0.545377\pi$
$878$	0	0
$879$	−7.61462	−0.256835
$880$	0	0
$881$	−41.5335	−1.39930	−0.699649	−	0.714486i	$-0.746660\pi$
$881$	−41.5335	−1.39930	−0.699649	+	0.714486i	$0.746660\pi$
$882$	0	0
$883$	56.4753	1.90054	0.950272	−	0.311422i	$-0.100805\pi$
$883$	56.4753	1.90054	0.950272	+	0.311422i	$0.100805\pi$
$884$	0	0
$885$	−16.4986	−0.554593
$886$	0	0
$887$	−34.5795	−1.16107	−0.580533	−	0.814237i	$-0.697156\pi$
$887$	−34.5795	−1.16107	−0.580533	+	0.814237i	$0.697156\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−11.1383	−0.373146
$892$	0	0
$893$	15.7673	0.527632
$894$	0	0
$895$	12.4962	0.417701
$896$	0	0
$897$	8.57753	0.286395
$898$	0	0
$899$	−67.0702	−2.23692
$900$	0	0
$901$	21.1712	0.705315
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.1723	0.504344
$906$	0	0
$907$	−8.40284	−0.279012	−0.139506	−	0.990221i	$-0.544551\pi$
$907$	−8.40284	−0.279012	−0.139506	+	0.990221i	$0.544551\pi$
$908$	0	0
$909$	18.1592	0.602303
$910$	0	0
$911$	0.593689	0.0196698	0.00983489	−	0.999952i	$-0.496869\pi$
$911$	0.593689	0.0196698	0.00983489	+	0.999952i	$0.496869\pi$
$912$	0	0
$913$	−9.23163	−0.305522
$914$	0	0
$915$	−6.05278	−0.200099
$916$	0	0
$917$	0	0
$918$	0	0
$919$	16.3370	0.538907	0.269454	−	0.963013i	$-0.413157\pi$
$919$	16.3370	0.538907	0.269454	+	0.963013i	$0.413157\pi$
$920$	0	0
$921$	14.3239	0.471988
$922$	0	0
$923$	−7.73128	−0.254478
$924$	0	0
$925$	−27.8792	−0.916662
$926$	0	0
$927$	11.4303	0.375421
$928$	0	0
$929$	−8.90120	−0.292039	−0.146019	−	0.989282i	$-0.546646\pi$
$929$	−8.90120	−0.292039	−0.146019	+	0.989282i	$0.546646\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−25.5710	−0.837156
$934$	0	0
$935$	1.80131	0.0589091
$936$	0	0
$937$	−45.2360	−1.47780	−0.738898	−	0.673817i	$-0.764653\pi$
$937$	−45.2360	−1.47780	−0.738898	+	0.673817i	$0.764653\pi$
$938$	0	0
$939$	−59.5795	−1.94430
$940$	0	0
$941$	6.93305	0.226011	0.113005	−	0.993594i	$-0.463952\pi$
$941$	6.93305	0.226011	0.113005	+	0.993594i	$0.463952\pi$
$942$	0	0
$943$	16.3204	0.531466
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.7433	−0.674066	−0.337033	−	0.941493i	$-0.609423\pi$
$947$	−20.7433	−0.674066	−0.337033	+	0.941493i	$0.609423\pi$
$948$	0	0
$949$	28.8092	0.935185
$950$	0	0
$951$	−8.49073	−0.275331
$952$	0	0
$953$	20.2076	0.654589	0.327295	−	0.944922i	$-0.393863\pi$
$953$	20.2076	0.654589	0.327295	+	0.944922i	$0.393863\pi$
$954$	0	0
$955$	7.07480	0.228935
$956$	0	0
$957$	22.9330	0.741320
$958$	0	0
$959$	0	0
$960$	0	0
$961$	10.3490	0.333838
$962$	0	0
$963$	−20.3579	−0.656024
$964$	0	0
$965$	2.29728	0.0739520
$966$	0	0
$967$	7.98254	0.256701	0.128351	−	0.991729i	$-0.459032\pi$
$967$	7.98254	0.256701	0.128351	+	0.991729i	$0.459032\pi$
$968$	0	0
$969$	34.6674	1.11368
$970$	0	0
$971$	−13.7793	−0.442199	−0.221099	−	0.975251i	$-0.570964\pi$
$971$	−13.7793	−0.442199	−0.221099	+	0.975251i	$0.570964\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	18.2029	0.582958
$976$	0	0
$977$	12.0209	0.384584	0.192292	−	0.981338i	$-0.438408\pi$
$977$	12.0209	0.384584	0.192292	+	0.981338i	$0.438408\pi$
$978$	0	0
$979$	−0.364448	−0.0116478
$980$	0	0
$981$	−26.2371	−0.837686
$982$	0	0
$983$	−44.6543	−1.42425	−0.712126	−	0.702052i	$-0.752267\pi$
$983$	−44.6543	−1.42425	−0.712126	+	0.702052i	$0.752267\pi$
$984$	0	0
$985$	1.53696	0.0489718
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10.5411	−0.335188
$990$	0	0
$991$	22.9660	0.729538	0.364769	−	0.931098i	$-0.381148\pi$
$991$	22.9660	0.729538	0.364769	+	0.931098i	$0.381148\pi$
$992$	0	0
$993$	13.5226	0.429126
$994$	0	0
$995$	−11.7553	−0.372668
$996$	0	0
$997$	−1.34898	−0.0427225	−0.0213612	−	0.999772i	$-0.506800\pi$
$997$	−1.34898	−0.0427225	−0.0213612	+	0.999772i	$0.506800\pi$
$998$	0	0
$999$	15.5477	0.491906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.ck.1.1		3
4.3	odd	2		539.2.a.i.1.3		3
7.3	odd	6		1232.2.q.k.177.1		6
7.5	odd	6		1232.2.q.k.529.1		6
7.6	odd	2		8624.2.a.cl.1.3		3
12.11	even	2		4851.2.a.bn.1.1		3
28.3	even	6		77.2.e.b.23.1	✓	6
28.11	odd	6		539.2.e.l.177.1		6
28.19	even	6		77.2.e.b.67.1	yes	6
28.23	odd	6		539.2.e.l.67.1		6
28.27	even	2		539.2.a.h.1.3		3
44.43	even	2		5929.2.a.w.1.1		3
84.47	odd	6		693.2.i.g.298.3		6
84.59	odd	6		693.2.i.g.100.3		6
84.83	odd	2		4851.2.a.bo.1.1		3
308.3	even	30		847.2.n.e.9.3		24
308.19	odd	30		847.2.n.d.130.3		24
308.31	even	30		847.2.n.e.807.3		24
308.47	even	30		847.2.n.e.130.1		24
308.59	even	30		847.2.n.e.632.1		24
308.75	even	30		847.2.n.e.81.1		24
308.87	odd	6		847.2.e.d.485.3		6
308.103	even	30		847.2.n.e.753.3		24
308.115	even	30		847.2.n.e.366.1		24
308.131	odd	6		847.2.e.d.606.3		6
308.159	even	30		847.2.n.e.487.3		24
308.171	odd	30		847.2.n.d.366.3		24
308.215	odd	30		847.2.n.d.487.1		24
308.227	odd	30		847.2.n.d.632.3		24
308.255	odd	30		847.2.n.d.807.1		24
308.271	odd	30		847.2.n.d.753.1		24
308.283	odd	30		847.2.n.d.9.1		24
308.299	odd	30		847.2.n.d.81.3		24
308.307	odd	2		5929.2.a.v.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.e.b.23.1	✓	6	28.3	even	6
77.2.e.b.67.1	yes	6	28.19	even	6
539.2.a.h.1.3		3	28.27	even	2
539.2.a.i.1.3		3	4.3	odd	2
539.2.e.l.67.1		6	28.23	odd	6
539.2.e.l.177.1		6	28.11	odd	6
693.2.i.g.100.3		6	84.59	odd	6
693.2.i.g.298.3		6	84.47	odd	6
847.2.e.d.485.3		6	308.87	odd	6
847.2.e.d.606.3		6	308.131	odd	6
847.2.n.d.9.1		24	308.283	odd	30
847.2.n.d.81.3		24	308.299	odd	30
847.2.n.d.130.3		24	308.19	odd	30
847.2.n.d.366.3		24	308.171	odd	30
847.2.n.d.487.1		24	308.215	odd	30
847.2.n.d.632.3		24	308.227	odd	30
847.2.n.d.753.1		24	308.271	odd	30
847.2.n.d.807.1		24	308.255	odd	30
847.2.n.e.9.3		24	308.3	even	30
847.2.n.e.81.1		24	308.75	even	30
847.2.n.e.130.1		24	308.47	even	30
847.2.n.e.366.1		24	308.115	even	30
847.2.n.e.487.3		24	308.159	even	30
847.2.n.e.632.1		24	308.59	even	30
847.2.n.e.753.3		24	308.103	even	30
847.2.n.e.807.3		24	308.31	even	30
1232.2.q.k.177.1		6	7.3	odd	6
1232.2.q.k.529.1		6	7.5	odd	6
4851.2.a.bn.1.1		3	12.11	even	2
4851.2.a.bo.1.1		3	84.83	odd	2
5929.2.a.v.1.1		3	308.307	odd	2
5929.2.a.w.1.1		3	44.43	even	2
8624.2.a.ck.1.1		3	1.1	even	1	trivial
8624.2.a.cl.1.3		3	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.19869 q^{3} +0.635552 q^{5} +1.83424 q^{9} +O(q^{10})\)	\(q-2.19869 q^{3} +0.635552 q^{5} +1.83424 q^{9} +1.00000 q^{11} +1.80131 q^{13} -1.39738 q^{15} +2.83424 q^{17} -5.56314 q^{19} -2.16576 q^{23} -4.59607 q^{25} +2.56314 q^{27} -10.4303 q^{29} +6.43032 q^{31} -2.19869 q^{33} +6.06587 q^{37} -3.96052 q^{39} -7.53566 q^{41} +4.86718 q^{43} +1.16576 q^{45} -2.83424 q^{47} -6.23163 q^{51} +7.46980 q^{53} +0.635552 q^{55} +12.2316 q^{57} +11.8068 q^{59} +4.33151 q^{61} +1.14483 q^{65} +1.60262 q^{67} +4.76183 q^{69} -4.29204 q^{71} +15.9935 q^{73} +10.1053 q^{75} -4.76183 q^{79} -11.1383 q^{81} -9.23163 q^{83} +1.80131 q^{85} +22.9330 q^{87} -0.364448 q^{89} -14.1383 q^{93} -3.53566 q^{95} +2.59607 q^{97} +1.83424 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 2 q^{5}+O(q^{10}) \) 3 * q - q^3 + 2 * q^5	\( 3 q - q^{3} + 2 q^{5} + 3 q^{11} + 11 q^{13} + 7 q^{15} + 3 q^{17} - 11 q^{19} - 12 q^{23} + 3 q^{25} + 2 q^{27} - 9 q^{29} - 3 q^{31} - q^{33} - 4 q^{37} + 5 q^{39} + 5 q^{41} - 2 q^{43} + 9 q^{45} - 3 q^{47} - 2 q^{51} + 17 q^{53} + 2 q^{55} + 20 q^{57} + 8 q^{59} + 24 q^{61} + 15 q^{65} + 16 q^{67} + 3 q^{69} - 7 q^{71} + 20 q^{73} + 25 q^{75} - 3 q^{79} - 17 q^{81} - 11 q^{83} + 11 q^{85} + 30 q^{87} - q^{89} - 26 q^{93} + 17 q^{95} - 9 q^{97}+O(q^{100}) \) 3 * q - q^3 + 2 * q^5 + 3 * q^11 + 11 * q^13 + 7 * q^15 + 3 * q^17 - 11 * q^19 - 12 * q^23 + 3 * q^25 + 2 * q^27 - 9 * q^29 - 3 * q^31 - q^33 - 4 * q^37 + 5 * q^39 + 5 * q^41 - 2 * q^43 + 9 * q^45 - 3 * q^47 - 2 * q^51 + 17 * q^53 + 2 * q^55 + 20 * q^57 + 8 * q^59 + 24 * q^61 + 15 * q^65 + 16 * q^67 + 3 * q^69 - 7 * q^71 + 20 * q^73 + 25 * q^75 - 3 * q^79 - 17 * q^81 - 11 * q^83 + 11 * q^85 + 30 * q^87 - q^89 - 26 * q^93 + 17 * q^95 - 9 * q^97

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