LMFDB - Embedded newform 3840.2.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(1,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, 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("3840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.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:	$30.6625543762$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	3840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +0.941367 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +0.941367 q^{7} +1.00000 q^{9} -4.49828 q^{11} +5.55691 q^{13} +1.00000 q^{15} +7.55691 q^{17} +1.05863 q^{19} +0.941367 q^{21} -1.05863 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} -3.55691 q^{31} -4.49828 q^{33} +0.941367 q^{35} -7.43965 q^{37} +5.55691 q^{39} +3.88273 q^{41} +1.88273 q^{43} +1.00000 q^{45} +10.0552 q^{47} -6.11383 q^{49} +7.55691 q^{51} +2.00000 q^{53} -4.49828 q^{55} +1.05863 q^{57} +8.49828 q^{59} -8.99656 q^{61} +0.941367 q^{63} +5.55691 q^{65} -4.00000 q^{67} -1.05863 q^{69} -12.9966 q^{71} +6.00000 q^{73} +1.00000 q^{75} -4.23453 q^{77} -11.5569 q^{79} +1.00000 q^{81} +5.88273 q^{83} +7.55691 q^{85} +2.00000 q^{87} +4.11727 q^{89} +5.23109 q^{91} -3.55691 q^{93} +1.05863 q^{95} +17.1138 q^{97} -4.49828 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9} + 4 q^{11} + 3 q^{15} + 6 q^{17} + 4 q^{19} + 2 q^{21} - 4 q^{23} + 3 q^{25} + 3 q^{27} + 6 q^{29} + 6 q^{31} + 4 q^{33} + 2 q^{35} - 4 q^{37} + 10 q^{41} + 4 q^{43} + 3 q^{45} - 4 q^{47} + 15 q^{49} + 6 q^{51} + 6 q^{53} + 4 q^{55} + 4 q^{57} + 8 q^{59} + 8 q^{61} + 2 q^{63} - 12 q^{67} - 4 q^{69} - 4 q^{71} + 18 q^{73} + 3 q^{75} - 16 q^{77} - 18 q^{79} + 3 q^{81} + 16 q^{83} + 6 q^{85} + 6 q^{87} + 14 q^{89} - 16 q^{91} + 6 q^{93} + 4 q^{95} + 18 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 3 * q^9 + 4 * q^11 + 3 * q^15 + 6 * q^17 + 4 * q^19 + 2 * q^21 - 4 * q^23 + 3 * q^25 + 3 * q^27 + 6 * q^29 + 6 * q^31 + 4 * q^33 + 2 * q^35 - 4 * q^37 + 10 * q^41 + 4 * q^43 + 3 * q^45 - 4 * q^47 + 15 * q^49 + 6 * q^51 + 6 * q^53 + 4 * q^55 + 4 * q^57 + 8 * q^59 + 8 * q^61 + 2 * q^63 - 12 * q^67 - 4 * q^69 - 4 * q^71 + 18 * q^73 + 3 * q^75 - 16 * q^77 - 18 * q^79 + 3 * q^81 + 16 * q^83 + 6 * q^85 + 6 * q^87 + 14 * q^89 - 16 * q^91 + 6 * q^93 + 4 * q^95 + 18 * q^97 + 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.941367	0.355803	0.177902	−	0.984048i	$-0.443069\pi$
$7$	0.941367	0.355803	0.177902	+	0.984048i	$0.443069\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.49828	−1.35628	−0.678141	−	0.734931i	$-0.737214\pi$
$11$	−4.49828	−1.35628	−0.678141	+	0.734931i	$0.737214\pi$
$12$	0	0
$13$	5.55691	1.54121	0.770605	−	0.637313i	$-0.219954\pi$
$13$	5.55691	1.54121	0.770605	+	0.637313i	$0.219954\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	7.55691	1.83282	0.916410	−	0.400240i	$-0.131073\pi$
$17$	7.55691	1.83282	0.916410	+	0.400240i	$0.131073\pi$
$18$	0	0
$19$	1.05863	0.242867	0.121434	−	0.992600i	$-0.461251\pi$
$19$	1.05863	0.242867	0.121434	+	0.992600i	$0.461251\pi$
$20$	0	0
$21$	0.941367	0.205423
$22$	0	0
$23$	−1.05863	−0.220740	−0.110370	−	0.993891i	$-0.535204\pi$
$23$	−1.05863	−0.220740	−0.110370	+	0.993891i	$0.535204\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−3.55691	−0.638841	−0.319420	−	0.947613i	$-0.603488\pi$
$31$	−3.55691	−0.638841	−0.319420	+	0.947613i	$0.603488\pi$
$32$	0	0
$33$	−4.49828	−0.783050
$34$	0	0
$35$	0.941367	0.159120
$36$	0	0
$37$	−7.43965	−1.22307	−0.611535	−	0.791217i	$-0.709448\pi$
$37$	−7.43965	−1.22307	−0.611535	+	0.791217i	$0.709448\pi$
$38$	0	0
$39$	5.55691	0.889818
$40$	0	0
$41$	3.88273	0.606381	0.303191	−	0.952930i	$-0.401948\pi$
$41$	3.88273	0.606381	0.303191	+	0.952930i	$0.401948\pi$
$42$	0	0
$43$	1.88273	0.287114	0.143557	−	0.989642i	$-0.454146\pi$
$43$	1.88273	0.287114	0.143557	+	0.989642i	$0.454146\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	10.0552	1.46670	0.733350	−	0.679851i	$-0.237955\pi$
$47$	10.0552	1.46670	0.733350	+	0.679851i	$0.237955\pi$
$48$	0	0
$49$	−6.11383	−0.873404
$50$	0	0
$51$	7.55691	1.05818
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−4.49828	−0.606548
$56$	0	0
$57$	1.05863	0.140219
$58$	0	0
$59$	8.49828	1.10638	0.553191	−	0.833054i	$-0.313410\pi$
$59$	8.49828	1.10638	0.553191	+	0.833054i	$0.313410\pi$
$60$	0	0
$61$	−8.99656	−1.15189	−0.575946	−	0.817488i	$-0.695366\pi$
$61$	−8.99656	−1.15189	−0.575946	+	0.817488i	$0.695366\pi$
$62$	0	0
$63$	0.941367	0.118601
$64$	0	0
$65$	5.55691	0.689250
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−1.05863	−0.127444
$70$	0	0
$71$	−12.9966	−1.54241	−0.771204	−	0.636588i	$-0.780345\pi$
$71$	−12.9966	−1.54241	−0.771204	+	0.636588i	$0.780345\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−4.23453	−0.482570
$78$	0	0
$79$	−11.5569	−1.30025	−0.650127	−	0.759825i	$-0.725284\pi$
$79$	−11.5569	−1.30025	−0.650127	+	0.759825i	$0.725284\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.88273	0.645714	0.322857	−	0.946448i	$-0.395357\pi$
$83$	5.88273	0.645714	0.322857	+	0.946448i	$0.395357\pi$
$84$	0	0
$85$	7.55691	0.819662
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	4.11727	0.436429	0.218215	−	0.975901i	$-0.429977\pi$
$89$	4.11727	0.436429	0.218215	+	0.975901i	$0.429977\pi$
$90$	0	0
$91$	5.23109	0.548368
$92$	0	0
$93$	−3.55691	−0.368835
$94$	0	0
$95$	1.05863	0.108613
$96$	0	0
$97$	17.1138	1.73765	0.868823	−	0.495123i	$-0.164877\pi$
$97$	17.1138	1.73765	0.868823	+	0.495123i	$0.164877\pi$
$98$	0	0
$99$	−4.49828	−0.452094
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	10.1725	1.00232	0.501161	−	0.865354i	$-0.332906\pi$
$103$	10.1725	1.00232	0.501161	+	0.865354i	$0.332906\pi$
$104$	0	0
$105$	0.941367	0.0918680
$106$	0	0
$107$	17.2311	1.66579	0.832896	−	0.553429i	$-0.186681\pi$
$107$	17.2311	1.66579	0.832896	+	0.553429i	$0.186681\pi$
$108$	0	0
$109$	−1.88273	−0.180333	−0.0901666	−	0.995927i	$-0.528740\pi$
$109$	−1.88273	−0.180333	−0.0901666	+	0.995927i	$0.528740\pi$
$110$	0	0
$111$	−7.43965	−0.706140
$112$	0	0
$113$	15.3224	1.44141	0.720704	−	0.693243i	$-0.243819\pi$
$113$	15.3224	1.44141	0.720704	+	0.693243i	$0.243819\pi$
$114$	0	0
$115$	−1.05863	−0.0987181
$116$	0	0
$117$	5.55691	0.513737
$118$	0	0
$119$	7.11383	0.652124
$120$	0	0
$121$	9.23453	0.839503
$122$	0	0
$123$	3.88273	0.350094
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	18.1725	1.61255	0.806273	−	0.591544i	$-0.201481\pi$
$127$	18.1725	1.61255	0.806273	+	0.591544i	$0.201481\pi$
$128$	0	0
$129$	1.88273	0.165765
$130$	0	0
$131$	−6.38101	−0.557512	−0.278756	−	0.960362i	$-0.589922\pi$
$131$	−6.38101	−0.557512	−0.278756	+	0.960362i	$0.589922\pi$
$132$	0	0
$133$	0.996562	0.0864129
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	4.44309	0.379598	0.189799	−	0.981823i	$-0.439216\pi$
$137$	4.44309	0.379598	0.189799	+	0.981823i	$0.439216\pi$
$138$	0	0
$139$	−20.1725	−1.71101	−0.855503	−	0.517798i	$-0.826752\pi$
$139$	−20.1725	−1.71101	−0.855503	+	0.517798i	$0.826752\pi$
$140$	0	0
$141$	10.0552	0.846800
$142$	0	0
$143$	−24.9966	−2.09032
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	−6.11383	−0.504260
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	9.67418	0.787274	0.393637	−	0.919266i	$-0.371217\pi$
$151$	9.67418	0.787274	0.393637	+	0.919266i	$0.371217\pi$
$152$	0	0
$153$	7.55691	0.610940
$154$	0	0
$155$	−3.55691	−0.285698
$156$	0	0
$157$	−4.32582	−0.345238	−0.172619	−	0.984989i	$-0.555223\pi$
$157$	−4.32582	−0.345238	−0.172619	+	0.984989i	$0.555223\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	−0.996562	−0.0785401
$162$	0	0
$163$	−6.11727	−0.479141	−0.239571	−	0.970879i	$-0.577007\pi$
$163$	−6.11727	−0.479141	−0.239571	+	0.970879i	$0.577007\pi$
$164$	0	0
$165$	−4.49828	−0.350191
$166$	0	0
$167$	−6.05520	−0.468565	−0.234283	−	0.972169i	$-0.575274\pi$
$167$	−6.05520	−0.468565	−0.234283	+	0.972169i	$0.575274\pi$
$168$	0	0
$169$	17.8793	1.37533
$170$	0	0
$171$	1.05863	0.0809557
$172$	0	0
$173$	−16.8793	−1.28331	−0.641655	−	0.766994i	$-0.721752\pi$
$173$	−16.8793	−1.28331	−0.641655	+	0.766994i	$0.721752\pi$
$174$	0	0
$175$	0.941367	0.0711606
$176$	0	0
$177$	8.49828	0.638770
$178$	0	0
$179$	10.6155	0.793443	0.396722	−	0.917939i	$-0.370148\pi$
$179$	10.6155	0.793443	0.396722	+	0.917939i	$0.370148\pi$
$180$	0	0
$181$	14.1173	1.04933	0.524664	−	0.851309i	$-0.324191\pi$
$181$	14.1173	1.04933	0.524664	+	0.851309i	$0.324191\pi$
$182$	0	0
$183$	−8.99656	−0.665045
$184$	0	0
$185$	−7.43965	−0.546974
$186$	0	0
$187$	−33.9931	−2.48582
$188$	0	0
$189$	0.941367	0.0684744
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−4.87930	−0.351219	−0.175610	−	0.984460i	$-0.556190\pi$
$193$	−4.87930	−0.351219	−0.175610	+	0.984460i	$0.556190\pi$
$194$	0	0
$195$	5.55691	0.397939
$196$	0	0
$197$	−2.88617	−0.205631	−0.102816	−	0.994700i	$-0.532785\pi$
$197$	−2.88617	−0.205631	−0.102816	+	0.994700i	$0.532785\pi$
$198$	0	0
$199$	−17.6742	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$199$	−17.6742	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	1.88273	0.132142
$204$	0	0
$205$	3.88273	0.271182
$206$	0	0
$207$	−1.05863	−0.0735801
$208$	0	0
$209$	−4.76203	−0.329396
$210$	0	0
$211$	−23.9379	−1.64795	−0.823977	−	0.566623i	$-0.808250\pi$
$211$	−23.9379	−1.64795	−0.823977	+	0.566623i	$0.808250\pi$
$212$	0	0
$213$	−12.9966	−0.890510
$214$	0	0
$215$	1.88273	0.128401
$216$	0	0
$217$	−3.34836	−0.227302
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	41.9931	2.82476
$222$	0	0
$223$	24.0552	1.61086	0.805428	−	0.592694i	$-0.201936\pi$
$223$	24.0552	1.61086	0.805428	+	0.592694i	$0.201936\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	11.1138	0.737651	0.368825	−	0.929499i	$-0.379760\pi$
$227$	11.1138	0.737651	0.368825	+	0.929499i	$0.379760\pi$
$228$	0	0
$229$	−17.2311	−1.13866	−0.569331	−	0.822108i	$-0.692798\pi$
$229$	−17.2311	−1.13866	−0.569331	+	0.822108i	$0.692798\pi$
$230$	0	0
$231$	−4.23453	−0.278612
$232$	0	0
$233$	8.44309	0.553125	0.276562	−	0.960996i	$-0.410805\pi$
$233$	8.44309	0.553125	0.276562	+	0.960996i	$0.410805\pi$
$234$	0	0
$235$	10.0552	0.655929
$236$	0	0
$237$	−11.5569	−0.750702
$238$	0	0
$239$	−10.1173	−0.654432	−0.327216	−	0.944950i	$-0.606110\pi$
$239$	−10.1173	−0.654432	−0.327216	+	0.944950i	$0.606110\pi$
$240$	0	0
$241$	16.8793	1.08729	0.543646	−	0.839315i	$-0.317044\pi$
$241$	16.8793	1.08729	0.543646	+	0.839315i	$0.317044\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−6.11383	−0.390598
$246$	0	0
$247$	5.88273	0.374309
$248$	0	0
$249$	5.88273	0.372803
$250$	0	0
$251$	−11.8466	−0.747753	−0.373877	−	0.927478i	$-0.621972\pi$
$251$	−11.8466	−0.747753	−0.373877	+	0.927478i	$0.621972\pi$
$252$	0	0
$253$	4.76203	0.299386
$254$	0	0
$255$	7.55691	0.473232
$256$	0	0
$257$	−10.6707	−0.665623	−0.332811	−	0.942993i	$-0.607997\pi$
$257$	−10.6707	−0.665623	−0.332811	+	0.942993i	$0.607997\pi$
$258$	0	0
$259$	−7.00344	−0.435172
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−1.94480	−0.119922	−0.0599609	−	0.998201i	$-0.519098\pi$
$263$	−1.94480	−0.119922	−0.0599609	+	0.998201i	$0.519098\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	4.11727	0.251973
$268$	0	0
$269$	−9.76547	−0.595411	−0.297706	−	0.954658i	$-0.596221\pi$
$269$	−9.76547	−0.595411	−0.297706	+	0.954658i	$0.596221\pi$
$270$	0	0
$271$	−3.44652	−0.209361	−0.104681	−	0.994506i	$-0.533382\pi$
$271$	−3.44652	−0.209361	−0.104681	+	0.994506i	$0.533382\pi$
$272$	0	0
$273$	5.23109	0.316600
$274$	0	0
$275$	−4.49828	−0.271257
$276$	0	0
$277$	−18.7880	−1.12886	−0.564431	−	0.825480i	$-0.690904\pi$
$277$	−18.7880	−1.12886	−0.564431	+	0.825480i	$0.690904\pi$
$278$	0	0
$279$	−3.55691	−0.212947
$280$	0	0
$281$	16.8793	1.00693	0.503467	−	0.864014i	$-0.332057\pi$
$281$	16.8793	1.00693	0.503467	+	0.864014i	$0.332057\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	1.05863	0.0627080
$286$	0	0
$287$	3.65508	0.215752
$288$	0	0
$289$	40.1070	2.35923
$290$	0	0
$291$	17.1138	1.00323
$292$	0	0
$293$	−20.2277	−1.18171	−0.590856	−	0.806777i	$-0.701210\pi$
$293$	−20.2277	−1.18171	−0.590856	+	0.806777i	$0.701210\pi$
$294$	0	0
$295$	8.49828	0.494789
$296$	0	0
$297$	−4.49828	−0.261017
$298$	0	0
$299$	−5.88273	−0.340207
$300$	0	0
$301$	1.77234	0.102156
$302$	0	0
$303$	2.00000	0.114897
$304$	0	0
$305$	−8.99656	−0.515142
$306$	0	0
$307$	8.11039	0.462884	0.231442	−	0.972849i	$-0.425656\pi$
$307$	8.11039	0.462884	0.231442	+	0.972849i	$0.425656\pi$
$308$	0	0
$309$	10.1725	0.578691
$310$	0	0
$311$	31.8759	1.80751	0.903757	−	0.428046i	$-0.140798\pi$
$311$	31.8759	1.80751	0.903757	+	0.428046i	$0.140798\pi$
$312$	0	0
$313$	5.11383	0.289051	0.144525	−	0.989501i	$-0.453834\pi$
$313$	5.11383	0.289051	0.144525	+	0.989501i	$0.453834\pi$
$314$	0	0
$315$	0.941367	0.0530400
$316$	0	0
$317$	−24.6448	−1.38419	−0.692094	−	0.721807i	$-0.743312\pi$
$317$	−24.6448	−1.38419	−0.692094	+	0.721807i	$0.743312\pi$
$318$	0	0
$319$	−8.99656	−0.503711
$320$	0	0
$321$	17.2311	0.961746
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	5.55691	0.308242
$326$	0	0
$327$	−1.88273	−0.104115
$328$	0	0
$329$	9.46563	0.521857
$330$	0	0
$331$	11.0518	0.607460	0.303730	−	0.952758i	$-0.401768\pi$
$331$	11.0518	0.607460	0.303730	+	0.952758i	$0.401768\pi$
$332$	0	0
$333$	−7.43965	−0.407690
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−19.9931	−1.08909	−0.544547	−	0.838730i	$-0.683299\pi$
$337$	−19.9931	−1.08909	−0.544547	+	0.838730i	$0.683299\pi$
$338$	0	0
$339$	15.3224	0.832198
$340$	0	0
$341$	16.0000	0.866449
$342$	0	0
$343$	−12.3449	−0.666563
$344$	0	0
$345$	−1.05863	−0.0569949
$346$	0	0
$347$	6.87930	0.369300	0.184650	−	0.982804i	$-0.440885\pi$
$347$	6.87930	0.369300	0.184650	+	0.982804i	$0.440885\pi$
$348$	0	0
$349$	4.76203	0.254906	0.127453	−	0.991845i	$-0.459320\pi$
$349$	4.76203	0.254906	0.127453	+	0.991845i	$0.459320\pi$
$350$	0	0
$351$	5.55691	0.296606
$352$	0	0
$353$	−3.79145	−0.201798	−0.100899	−	0.994897i	$-0.532172\pi$
$353$	−3.79145	−0.201798	−0.100899	+	0.994897i	$0.532172\pi$
$354$	0	0
$355$	−12.9966	−0.689786
$356$	0	0
$357$	7.11383	0.376504
$358$	0	0
$359$	−12.9966	−0.685932	−0.342966	−	0.939348i	$-0.611432\pi$
$359$	−12.9966	−0.685932	−0.342966	+	0.939348i	$0.611432\pi$
$360$	0	0
$361$	−17.8793	−0.941016
$362$	0	0
$363$	9.23453	0.484687
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	−22.9345	−1.19717	−0.598585	−	0.801059i	$-0.704270\pi$
$367$	−22.9345	−1.19717	−0.598585	+	0.801059i	$0.704270\pi$
$368$	0	0
$369$	3.88273	0.202127
$370$	0	0
$371$	1.88273	0.0977467
$372$	0	0
$373$	15.4396	0.799435	0.399717	−	0.916638i	$-0.369108\pi$
$373$	15.4396	0.799435	0.399717	+	0.916638i	$0.369108\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	11.1138	0.572391
$378$	0	0
$379$	6.28973	0.323082	0.161541	−	0.986866i	$-0.448354\pi$
$379$	6.28973	0.323082	0.161541	+	0.986866i	$0.448354\pi$
$380$	0	0
$381$	18.1725	0.931003
$382$	0	0
$383$	−2.94137	−0.150297	−0.0751484	−	0.997172i	$-0.523943\pi$
$383$	−2.94137	−0.150297	−0.0751484	+	0.997172i	$0.523943\pi$
$384$	0	0
$385$	−4.23453	−0.215812
$386$	0	0
$387$	1.88273	0.0957047
$388$	0	0
$389$	12.2277	0.619967	0.309983	−	0.950742i	$-0.399676\pi$
$389$	12.2277	0.619967	0.309983	+	0.950742i	$0.399676\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	0	0
$393$	−6.38101	−0.321880
$394$	0	0
$395$	−11.5569	−0.581491
$396$	0	0
$397$	5.32238	0.267123	0.133561	−	0.991041i	$-0.457359\pi$
$397$	5.32238	0.267123	0.133561	+	0.991041i	$0.457359\pi$
$398$	0	0
$399$	0.996562	0.0498905
$400$	0	0
$401$	6.99656	0.349392	0.174696	−	0.984622i	$-0.444106\pi$
$401$	6.99656	0.349392	0.174696	+	0.984622i	$0.444106\pi$
$402$	0	0
$403$	−19.7655	−0.984588
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	33.4656	1.65883
$408$	0	0
$409$	−16.2277	−0.802406	−0.401203	−	0.915989i	$-0.631408\pi$
$409$	−16.2277	−0.802406	−0.401203	+	0.915989i	$0.631408\pi$
$410$	0	0
$411$	4.44309	0.219161
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	5.88273	0.288772
$416$	0	0
$417$	−20.1725	−0.987850
$418$	0	0
$419$	−15.6121	−0.762701	−0.381351	−	0.924430i	$-0.624541\pi$
$419$	−15.6121	−0.762701	−0.381351	+	0.924430i	$0.624541\pi$
$420$	0	0
$421$	33.2311	1.61958	0.809792	−	0.586717i	$-0.199580\pi$
$421$	33.2311	1.61958	0.809792	+	0.586717i	$0.199580\pi$
$422$	0	0
$423$	10.0552	0.488900
$424$	0	0
$425$	7.55691	0.366564
$426$	0	0
$427$	−8.46907	−0.409847
$428$	0	0
$429$	−24.9966	−1.20685
$430$	0	0
$431$	12.9966	0.626022	0.313011	−	0.949749i	$-0.398662\pi$
$431$	12.9966	0.626022	0.313011	+	0.949749i	$0.398662\pi$
$432$	0	0
$433$	20.2277	0.972079	0.486040	−	0.873937i	$-0.338441\pi$
$433$	20.2277	0.972079	0.486040	+	0.873937i	$0.338441\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	0	0
$437$	−1.12070	−0.0536106
$438$	0	0
$439$	−5.43965	−0.259620	−0.129810	−	0.991539i	$-0.541437\pi$
$439$	−5.43965	−0.259620	−0.129810	+	0.991539i	$0.541437\pi$
$440$	0	0
$441$	−6.11383	−0.291135
$442$	0	0
$443$	15.3484	0.729223	0.364611	−	0.931160i	$-0.381202\pi$
$443$	15.3484	0.729223	0.364611	+	0.931160i	$0.381202\pi$
$444$	0	0
$445$	4.11727	0.195177
$446$	0	0
$447$	−2.00000	−0.0945968
$448$	0	0
$449$	4.22766	0.199515	0.0997577	−	0.995012i	$-0.468193\pi$
$449$	4.22766	0.199515	0.0997577	+	0.995012i	$0.468193\pi$
$450$	0	0
$451$	−17.4656	−0.822424
$452$	0	0
$453$	9.67418	0.454533
$454$	0	0
$455$	5.23109	0.245238
$456$	0	0
$457$	−2.65164	−0.124038	−0.0620192	−	0.998075i	$-0.519754\pi$
$457$	−2.65164	−0.124038	−0.0620192	+	0.998075i	$0.519754\pi$
$458$	0	0
$459$	7.55691	0.352727
$460$	0	0
$461$	10.2345	0.476670	0.238335	−	0.971183i	$-0.423398\pi$
$461$	10.2345	0.476670	0.238335	+	0.971183i	$0.423398\pi$
$462$	0	0
$463$	−19.0586	−0.885730	−0.442865	−	0.896588i	$-0.646038\pi$
$463$	−19.0586	−0.885730	−0.442865	+	0.896588i	$0.646038\pi$
$464$	0	0
$465$	−3.55691	−0.164948
$466$	0	0
$467$	−4.11039	−0.190206	−0.0951031	−	0.995467i	$-0.530318\pi$
$467$	−4.11039	−0.190206	−0.0951031	+	0.995467i	$0.530318\pi$
$468$	0	0
$469$	−3.76547	−0.173873
$470$	0	0
$471$	−4.32582	−0.199323
$472$	0	0
$473$	−8.46907	−0.389408
$474$	0	0
$475$	1.05863	0.0485734
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	25.2311	1.15284	0.576419	−	0.817154i	$-0.304450\pi$
$479$	25.2311	1.15284	0.576419	+	0.817154i	$0.304450\pi$
$480$	0	0
$481$	−41.3415	−1.88501
$482$	0	0
$483$	−0.996562	−0.0453451
$484$	0	0
$485$	17.1138	0.777099
$486$	0	0
$487$	−21.9379	−0.994102	−0.497051	−	0.867721i	$-0.665584\pi$
$487$	−21.9379	−0.994102	−0.497051	+	0.867721i	$0.665584\pi$
$488$	0	0
$489$	−6.11727	−0.276632
$490$	0	0
$491$	7.50172	0.338548	0.169274	−	0.985569i	$-0.445858\pi$
$491$	7.50172	0.338548	0.169274	+	0.985569i	$0.445858\pi$
$492$	0	0
$493$	15.1138	0.680693
$494$	0	0
$495$	−4.49828	−0.202183
$496$	0	0
$497$	−12.2345	−0.548794
$498$	0	0
$499$	−29.1690	−1.30578	−0.652892	−	0.757451i	$-0.726445\pi$
$499$	−29.1690	−1.30578	−0.652892	+	0.757451i	$0.726445\pi$
$500$	0	0
$501$	−6.05520	−0.270526
$502$	0	0
$503$	−23.9379	−1.06734	−0.533670	−	0.845693i	$-0.679187\pi$
$503$	−23.9379	−1.06734	−0.533670	+	0.845693i	$0.679187\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	17.8793	0.794047
$508$	0	0
$509$	−28.6967	−1.27196	−0.635980	−	0.771706i	$-0.719404\pi$
$509$	−28.6967	−1.27196	−0.635980	+	0.771706i	$0.719404\pi$
$510$	0	0
$511$	5.64820	0.249862
$512$	0	0
$513$	1.05863	0.0467398
$514$	0	0
$515$	10.1725	0.448252
$516$	0	0
$517$	−45.2311	−1.98926
$518$	0	0
$519$	−16.8793	−0.740919
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	−25.7586	−1.12634	−0.563172	−	0.826340i	$-0.690419\pi$
$523$	−25.7586	−1.12634	−0.563172	+	0.826340i	$0.690419\pi$
$524$	0	0
$525$	0.941367	0.0410846
$526$	0	0
$527$	−26.8793	−1.17088
$528$	0	0
$529$	−21.8793	−0.951274
$530$	0	0
$531$	8.49828	0.368794
$532$	0	0
$533$	21.5760	0.934561
$534$	0	0
$535$	17.2311	0.744965
$536$	0	0
$537$	10.6155	0.458095
$538$	0	0
$539$	27.5017	1.18458
$540$	0	0
$541$	−12.3449	−0.530750	−0.265375	−	0.964145i	$-0.585496\pi$
$541$	−12.3449	−0.530750	−0.265375	+	0.964145i	$0.585496\pi$
$542$	0	0
$543$	14.1173	0.605830
$544$	0	0
$545$	−1.88273	−0.0806475
$546$	0	0
$547$	−19.8759	−0.849830	−0.424915	−	0.905233i	$-0.639696\pi$
$547$	−19.8759	−0.849830	−0.424915	+	0.905233i	$0.639696\pi$
$548$	0	0
$549$	−8.99656	−0.383964
$550$	0	0
$551$	2.11727	0.0901986
$552$	0	0
$553$	−10.8793	−0.462635
$554$	0	0
$555$	−7.43965	−0.315795
$556$	0	0
$557$	−3.12070	−0.132228	−0.0661142	−	0.997812i	$-0.521060\pi$
$557$	−3.12070	−0.132228	−0.0661142	+	0.997812i	$0.521060\pi$
$558$	0	0
$559$	10.4622	0.442503
$560$	0	0
$561$	−33.9931	−1.43519
$562$	0	0
$563$	−0.651639	−0.0274633	−0.0137317	−	0.999906i	$-0.504371\pi$
$563$	−0.651639	−0.0274633	−0.0137317	+	0.999906i	$0.504371\pi$
$564$	0	0
$565$	15.3224	0.644617
$566$	0	0
$567$	0.941367	0.0395337
$568$	0	0
$569$	−26.9966	−1.13175	−0.565877	−	0.824489i	$-0.691462\pi$
$569$	−26.9966	−1.13175	−0.565877	+	0.824489i	$0.691462\pi$
$570$	0	0
$571$	−14.9414	−0.625277	−0.312638	−	0.949872i	$-0.601213\pi$
$571$	−14.9414	−0.625277	−0.312638	+	0.949872i	$0.601213\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	−1.05863	−0.0441481
$576$	0	0
$577$	−8.87930	−0.369650	−0.184825	−	0.982771i	$-0.559172\pi$
$577$	−8.87930	−0.369650	−0.184825	+	0.982771i	$0.559172\pi$
$578$	0	0
$579$	−4.87930	−0.202777
$580$	0	0
$581$	5.53781	0.229747
$582$	0	0
$583$	−8.99656	−0.372600
$584$	0	0
$585$	5.55691	0.229750
$586$	0	0
$587$	−1.23109	−0.0508127	−0.0254064	−	0.999677i	$-0.508088\pi$
$587$	−1.23109	−0.0508127	−0.0254064	+	0.999677i	$0.508088\pi$
$588$	0	0
$589$	−3.76547	−0.155153
$590$	0	0
$591$	−2.88617	−0.118721
$592$	0	0
$593$	3.55691	0.146065	0.0730325	−	0.997330i	$-0.476732\pi$
$593$	3.55691	0.146065	0.0730325	+	0.997330i	$0.476732\pi$
$594$	0	0
$595$	7.11383	0.291639
$596$	0	0
$597$	−17.6742	−0.723356
$598$	0	0
$599$	−19.2242	−0.785480	−0.392740	−	0.919649i	$-0.628473\pi$
$599$	−19.2242	−0.785480	−0.392740	+	0.919649i	$0.628473\pi$
$600$	0	0
$601$	27.7586	1.13230	0.566148	−	0.824303i	$-0.308433\pi$
$601$	27.7586	1.13230	0.566148	+	0.824303i	$0.308433\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	9.23453	0.375437
$606$	0	0
$607$	−7.16902	−0.290982	−0.145491	−	0.989360i	$-0.546476\pi$
$607$	−7.16902	−0.290982	−0.145491	+	0.989360i	$0.546476\pi$
$608$	0	0
$609$	1.88273	0.0762922
$610$	0	0
$611$	55.8759	2.26050
$612$	0	0
$613$	9.55691	0.386000	0.193000	−	0.981199i	$-0.438178\pi$
$613$	9.55691	0.386000	0.193000	+	0.981199i	$0.438178\pi$
$614$	0	0
$615$	3.88273	0.156567
$616$	0	0
$617$	−1.32926	−0.0535139	−0.0267569	−	0.999642i	$-0.508518\pi$
$617$	−1.32926	−0.0535139	−0.0267569	+	0.999642i	$0.508518\pi$
$618$	0	0
$619$	−28.1725	−1.13235	−0.566173	−	0.824286i	$-0.691577\pi$
$619$	−28.1725	−1.13235	−0.566173	+	0.824286i	$0.691577\pi$
$620$	0	0
$621$	−1.05863	−0.0424815
$622$	0	0
$623$	3.87586	0.155283
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.76203	−0.190177
$628$	0	0
$629$	−56.2208	−2.24167
$630$	0	0
$631$	−23.3224	−0.928449	−0.464225	−	0.885717i	$-0.653667\pi$
$631$	−23.3224	−0.928449	−0.464225	+	0.885717i	$0.653667\pi$
$632$	0	0
$633$	−23.9379	−0.951447
$634$	0	0
$635$	18.1725	0.721152
$636$	0	0
$637$	−33.9740	−1.34610
$638$	0	0
$639$	−12.9966	−0.514136
$640$	0	0
$641$	27.1070	1.07066	0.535330	−	0.844643i	$-0.320187\pi$
$641$	27.1070	1.07066	0.535330	+	0.844643i	$0.320187\pi$
$642$	0	0
$643$	20.3449	0.802325	0.401163	−	0.916007i	$-0.368606\pi$
$643$	20.3449	0.802325	0.401163	+	0.916007i	$0.368606\pi$
$644$	0	0
$645$	1.88273	0.0741326
$646$	0	0
$647$	37.6965	1.48200	0.741002	−	0.671503i	$-0.234351\pi$
$647$	37.6965	1.48200	0.741002	+	0.671503i	$0.234351\pi$
$648$	0	0
$649$	−38.2277	−1.50057
$650$	0	0
$651$	−3.34836	−0.131233
$652$	0	0
$653$	8.64476	0.338296	0.169148	−	0.985591i	$-0.445898\pi$
$653$	8.64476	0.338296	0.169148	+	0.985591i	$0.445898\pi$
$654$	0	0
$655$	−6.38101	−0.249327
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	29.2603	1.13982	0.569910	−	0.821707i	$-0.306978\pi$
$659$	29.2603	1.13982	0.569910	+	0.821707i	$0.306978\pi$
$660$	0	0
$661$	28.7620	1.11871	0.559357	−	0.828927i	$-0.311048\pi$
$661$	28.7620	1.11871	0.559357	+	0.828927i	$0.311048\pi$
$662$	0	0
$663$	41.9931	1.63088
$664$	0	0
$665$	0.996562	0.0386450
$666$	0	0
$667$	−2.11727	−0.0819809
$668$	0	0
$669$	24.0552	0.930028
$670$	0	0
$671$	40.4691	1.56229
$672$	0	0
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−42.8724	−1.64772	−0.823860	−	0.566793i	$-0.808184\pi$
$677$	−42.8724	−1.64772	−0.823860	+	0.566793i	$0.808184\pi$
$678$	0	0
$679$	16.1104	0.618260
$680$	0	0
$681$	11.1138	0.425883
$682$	0	0
$683$	26.1173	0.999349	0.499675	−	0.866213i	$-0.333453\pi$
$683$	26.1173	0.999349	0.499675	+	0.866213i	$0.333453\pi$
$684$	0	0
$685$	4.44309	0.169762
$686$	0	0
$687$	−17.2311	−0.657407
$688$	0	0
$689$	11.1138	0.423403
$690$	0	0
$691$	−5.29317	−0.201362	−0.100681	−	0.994919i	$-0.532102\pi$
$691$	−5.29317	−0.201362	−0.100681	+	0.994919i	$0.532102\pi$
$692$	0	0
$693$	−4.23453	−0.160857
$694$	0	0
$695$	−20.1725	−0.765185
$696$	0	0
$697$	29.3415	1.11139
$698$	0	0
$699$	8.44309	0.319347
$700$	0	0
$701$	−7.99312	−0.301896	−0.150948	−	0.988542i	$-0.548233\pi$
$701$	−7.99312	−0.301896	−0.150948	+	0.988542i	$0.548233\pi$
$702$	0	0
$703$	−7.87586	−0.297044
$704$	0	0
$705$	10.0552	0.378701
$706$	0	0
$707$	1.88273	0.0708075
$708$	0	0
$709$	−28.9966	−1.08899	−0.544494	−	0.838764i	$-0.683278\pi$
$709$	−28.9966	−1.08899	−0.544494	+	0.838764i	$0.683278\pi$
$710$	0	0
$711$	−11.5569	−0.433418
$712$	0	0
$713$	3.76547	0.141018
$714$	0	0
$715$	−24.9966	−0.934818
$716$	0	0
$717$	−10.1173	−0.377836
$718$	0	0
$719$	−26.8793	−1.00243	−0.501214	−	0.865323i	$-0.667113\pi$
$719$	−26.8793	−1.00243	−0.501214	+	0.865323i	$0.667113\pi$
$720$	0	0
$721$	9.57602	0.356630
$722$	0	0
$723$	16.8793	0.627748
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	41.8138	1.55079	0.775394	−	0.631478i	$-0.217551\pi$
$727$	41.8138	1.55079	0.775394	+	0.631478i	$0.217551\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	14.2277	0.526229
$732$	0	0
$733$	−30.0844	−1.11119	−0.555597	−	0.831452i	$-0.687510\pi$
$733$	−30.0844	−1.11119	−0.555597	+	0.831452i	$0.687510\pi$
$734$	0	0
$735$	−6.11383	−0.225512
$736$	0	0
$737$	17.9931	0.662785
$738$	0	0
$739$	29.0449	1.06843	0.534217	−	0.845348i	$-0.320607\pi$
$739$	29.0449	1.06843	0.534217	+	0.845348i	$0.320607\pi$
$740$	0	0
$741$	5.88273	0.216108
$742$	0	0
$743$	−43.2863	−1.58802	−0.794010	−	0.607905i	$-0.792010\pi$
$743$	−43.2863	−1.58802	−0.794010	+	0.607905i	$0.792010\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	0	0
$747$	5.88273	0.215238
$748$	0	0
$749$	16.2208	0.592694
$750$	0	0
$751$	−41.7846	−1.52474	−0.762370	−	0.647141i	$-0.775964\pi$
$751$	−41.7846	−1.52474	−0.762370	+	0.647141i	$0.775964\pi$
$752$	0	0
$753$	−11.8466	−0.431716
$754$	0	0
$755$	9.67418	0.352079
$756$	0	0
$757$	−16.3258	−0.593372	−0.296686	−	0.954975i	$-0.595881\pi$
$757$	−16.3258	−0.593372	−0.296686	+	0.954975i	$0.595881\pi$
$758$	0	0
$759$	4.76203	0.172851
$760$	0	0
$761$	−50.2208	−1.82050	−0.910251	−	0.414057i	$-0.864111\pi$
$761$	−50.2208	−1.82050	−0.910251	+	0.414057i	$0.864111\pi$
$762$	0	0
$763$	−1.77234	−0.0641631
$764$	0	0
$765$	7.55691	0.273221
$766$	0	0
$767$	47.2242	1.70517
$768$	0	0
$769$	−31.3415	−1.13020	−0.565101	−	0.825021i	$-0.691163\pi$
$769$	−31.3415	−1.13020	−0.565101	+	0.825021i	$0.691163\pi$
$770$	0	0
$771$	−10.6707	−0.384297
$772$	0	0
$773$	−9.11383	−0.327802	−0.163901	−	0.986477i	$-0.552408\pi$
$773$	−9.11383	−0.327802	−0.163901	+	0.986477i	$0.552408\pi$
$774$	0	0
$775$	−3.55691	−0.127768
$776$	0	0
$777$	−7.00344	−0.251247
$778$	0	0
$779$	4.11039	0.147270
$780$	0	0
$781$	58.4622	2.09194
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	−4.32582	−0.154395
$786$	0	0
$787$	36.2208	1.29113	0.645566	−	0.763705i	$-0.276622\pi$
$787$	36.2208	1.29113	0.645566	+	0.763705i	$0.276622\pi$
$788$	0	0
$789$	−1.94480	−0.0692369
$790$	0	0
$791$	14.4240	0.512858
$792$	0	0
$793$	−49.9931	−1.77531
$794$	0	0
$795$	2.00000	0.0709327
$796$	0	0
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	0	0
$799$	75.9862	2.68820
$800$	0	0
$801$	4.11727	0.145476
$802$	0	0
$803$	−26.9897	−0.952445
$804$	0	0
$805$	−0.996562	−0.0351242
$806$	0	0
$807$	−9.76547	−0.343761
$808$	0	0
$809$	−47.5760	−1.67268	−0.836342	−	0.548208i	$-0.815310\pi$
$809$	−47.5760	−1.67268	−0.836342	+	0.548208i	$0.815310\pi$
$810$	0	0
$811$	−20.5174	−0.720463	−0.360231	−	0.932863i	$-0.617302\pi$
$811$	−20.5174	−0.720463	−0.360231	+	0.932863i	$0.617302\pi$
$812$	0	0
$813$	−3.44652	−0.120875
$814$	0	0
$815$	−6.11727	−0.214278
$816$	0	0
$817$	1.99312	0.0697306
$818$	0	0
$819$	5.23109	0.182789
$820$	0	0
$821$	44.4622	1.55174	0.775871	−	0.630892i	$-0.217311\pi$
$821$	44.4622	1.55174	0.775871	+	0.630892i	$0.217311\pi$
$822$	0	0
$823$	32.1656	1.12122	0.560611	−	0.828079i	$-0.310566\pi$
$823$	32.1656	1.12122	0.560611	+	0.828079i	$0.310566\pi$
$824$	0	0
$825$	−4.49828	−0.156610
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	−33.8827	−1.17680	−0.588398	−	0.808571i	$-0.700241\pi$
$829$	−33.8827	−1.17680	−0.588398	+	0.808571i	$0.700241\pi$
$830$	0	0
$831$	−18.7880	−0.651749
$832$	0	0
$833$	−46.2017	−1.60079
$834$	0	0
$835$	−6.05520	−0.209549
$836$	0	0
$837$	−3.55691	−0.122945
$838$	0	0
$839$	4.52750	0.156307	0.0781533	−	0.996941i	$-0.475098\pi$
$839$	4.52750	0.156307	0.0781533	+	0.996941i	$0.475098\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	16.8793	0.581354
$844$	0	0
$845$	17.8793	0.615066
$846$	0	0
$847$	8.69308	0.298698
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	7.87586	0.269981
$852$	0	0
$853$	−50.4293	−1.72667	−0.863334	−	0.504633i	$-0.831628\pi$
$853$	−50.4293	−1.72667	−0.863334	+	0.504633i	$0.831628\pi$
$854$	0	0
$855$	1.05863	0.0362045
$856$	0	0
$857$	−26.4362	−0.903044	−0.451522	−	0.892260i	$-0.649119\pi$
$857$	−26.4362	−0.903044	−0.451522	+	0.892260i	$0.649119\pi$
$858$	0	0
$859$	−0.406994	−0.0138865	−0.00694323	−	0.999976i	$-0.502210\pi$
$859$	−0.406994	−0.0138865	−0.00694323	+	0.999976i	$0.502210\pi$
$860$	0	0
$861$	3.65508	0.124565
$862$	0	0
$863$	29.9311	1.01886	0.509432	−	0.860511i	$-0.329855\pi$
$863$	29.9311	1.01886	0.509432	+	0.860511i	$0.329855\pi$
$864$	0	0
$865$	−16.8793	−0.573913
$866$	0	0
$867$	40.1070	1.36210
$868$	0	0
$869$	51.9862	1.76351
$870$	0	0
$871$	−22.2277	−0.753155
$872$	0	0
$873$	17.1138	0.579215
$874$	0	0
$875$	0.941367	0.0318240
$876$	0	0
$877$	11.2051	0.378370	0.189185	−	0.981941i	$-0.439415\pi$
$877$	11.2051	0.378370	0.189185	+	0.981941i	$0.439415\pi$
$878$	0	0
$879$	−20.2277	−0.682262
$880$	0	0
$881$	−48.3380	−1.62855	−0.814275	−	0.580479i	$-0.802865\pi$
$881$	−48.3380	−1.62855	−0.814275	+	0.580479i	$0.802865\pi$
$882$	0	0
$883$	−50.5726	−1.70190	−0.850951	−	0.525244i	$-0.823974\pi$
$883$	−50.5726	−1.70190	−0.850951	+	0.525244i	$0.823974\pi$
$884$	0	0
$885$	8.49828	0.285667
$886$	0	0
$887$	−48.0483	−1.61330	−0.806652	−	0.591026i	$-0.798723\pi$
$887$	−48.0483	−1.61330	−0.806652	+	0.591026i	$0.798723\pi$
$888$	0	0
$889$	17.1070	0.573749
$890$	0	0
$891$	−4.49828	−0.150698
$892$	0	0
$893$	10.6448	0.356213
$894$	0	0
$895$	10.6155	0.354839
$896$	0	0
$897$	−5.88273	−0.196419
$898$	0	0
$899$	−7.11383	−0.237259
$900$	0	0
$901$	15.1138	0.503515
$902$	0	0
$903$	1.77234	0.0589799
$904$	0	0
$905$	14.1173	0.469274
$906$	0	0
$907$	6.46219	0.214573	0.107287	−	0.994228i	$-0.465784\pi$
$907$	6.46219	0.214573	0.107287	+	0.994228i	$0.465784\pi$
$908$	0	0
$909$	2.00000	0.0663358
$910$	0	0
$911$	−50.3380	−1.66777	−0.833887	−	0.551935i	$-0.813890\pi$
$911$	−50.3380	−1.66777	−0.833887	+	0.551935i	$0.813890\pi$
$912$	0	0
$913$	−26.4622	−0.875771
$914$	0	0
$915$	−8.99656	−0.297417
$916$	0	0
$917$	−6.00688	−0.198365
$918$	0	0
$919$	46.4362	1.53179	0.765895	−	0.642966i	$-0.222296\pi$
$919$	46.4362	1.53179	0.765895	+	0.642966i	$0.222296\pi$
$920$	0	0
$921$	8.11039	0.267246
$922$	0	0
$923$	−72.2208	−2.37718
$924$	0	0
$925$	−7.43965	−0.244614
$926$	0	0
$927$	10.1725	0.334107
$928$	0	0
$929$	−35.9931	−1.18090	−0.590448	−	0.807076i	$-0.701049\pi$
$929$	−35.9931	−1.18090	−0.590448	+	0.807076i	$0.701049\pi$
$930$	0	0
$931$	−6.47230	−0.212121
$932$	0	0
$933$	31.8759	1.04357
$934$	0	0
$935$	−33.9931	−1.11169
$936$	0	0
$937$	−2.70360	−0.0883227	−0.0441613	−	0.999024i	$-0.514062\pi$
$937$	−2.70360	−0.0883227	−0.0441613	+	0.999024i	$0.514062\pi$
$938$	0	0
$939$	5.11383	0.166883
$940$	0	0
$941$	17.7655	0.579138	0.289569	−	0.957157i	$-0.406488\pi$
$941$	17.7655	0.579138	0.289569	+	0.957157i	$0.406488\pi$
$942$	0	0
$943$	−4.11039	−0.133853
$944$	0	0
$945$	0.941367	0.0306227
$946$	0	0
$947$	−26.2277	−0.852284	−0.426142	−	0.904656i	$-0.640128\pi$
$947$	−26.2277	−0.852284	−0.426142	+	0.904656i	$0.640128\pi$
$948$	0	0
$949$	33.3415	1.08231
$950$	0	0
$951$	−24.6448	−0.799161
$952$	0	0
$953$	−9.09472	−0.294607	−0.147304	−	0.989091i	$-0.547059\pi$
$953$	−9.09472	−0.294607	−0.147304	+	0.989091i	$0.547059\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	0	0
$957$	−8.99656	−0.290818
$958$	0	0
$959$	4.18257	0.135062
$960$	0	0
$961$	−18.3484	−0.591883
$962$	0	0
$963$	17.2311	0.555264
$964$	0	0
$965$	−4.87930	−0.157070
$966$	0	0
$967$	−7.47574	−0.240404	−0.120202	−	0.992749i	$-0.538354\pi$
$967$	−7.47574	−0.240404	−0.120202	+	0.992749i	$0.538354\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	41.0777	1.31825	0.659124	−	0.752035i	$-0.270927\pi$
$971$	41.0777	1.31825	0.659124	+	0.752035i	$0.270927\pi$
$972$	0	0
$973$	−18.9897	−0.608781
$974$	0	0
$975$	5.55691	0.177964
$976$	0	0
$977$	−4.20855	−0.134644	−0.0673218	−	0.997731i	$-0.521445\pi$
$977$	−4.20855	−0.134644	−0.0673218	+	0.997731i	$0.521445\pi$
$978$	0	0
$979$	−18.5206	−0.591922
$980$	0	0
$981$	−1.88273	−0.0601111
$982$	0	0
$983$	−8.35504	−0.266484	−0.133242	−	0.991084i	$-0.542539\pi$
$983$	−8.35504	−0.266484	−0.133242	+	0.991084i	$0.542539\pi$
$984$	0	0
$985$	−2.88617	−0.0919611
$986$	0	0
$987$	9.46563	0.301294
$988$	0	0
$989$	−1.99312	−0.0633777
$990$	0	0
$991$	13.9087	0.441825	0.220912	−	0.975294i	$-0.429097\pi$
$991$	13.9087	0.441825	0.220912	+	0.975294i	$0.429097\pi$
$992$	0	0
$993$	11.0518	0.350717
$994$	0	0
$995$	−17.6742	−0.560309
$996$	0	0
$997$	34.8984	1.10524	0.552622	−	0.833432i	$-0.313627\pi$
$997$	34.8984	1.10524	0.552622	+	0.833432i	$0.313627\pi$
$998$	0	0
$999$	−7.43965	−0.235380

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.a.br.1.2		3
4.3	odd	2		3840.2.a.bp.1.2		3
8.3	odd	2		3840.2.a.bq.1.2		3
8.5	even	2		3840.2.a.bo.1.2		3
16.3	odd	4		120.2.k.b.61.3	✓	6
16.5	even	4		480.2.k.b.241.2		6
16.11	odd	4		120.2.k.b.61.4	yes	6
16.13	even	4		480.2.k.b.241.5		6
48.5	odd	4		1440.2.k.f.721.2		6
48.11	even	4		360.2.k.f.181.3		6
48.29	odd	4		1440.2.k.f.721.5		6
48.35	even	4		360.2.k.f.181.4		6
80.3	even	4		600.2.d.f.349.1		6
80.13	odd	4		2400.2.d.e.49.4		6
80.19	odd	4		600.2.k.c.301.4		6
80.27	even	4		600.2.d.f.349.2		6
80.29	even	4		2400.2.k.c.1201.2		6
80.37	odd	4		2400.2.d.e.49.3		6
80.43	even	4		600.2.d.e.349.5		6
80.53	odd	4		2400.2.d.f.49.4		6
80.59	odd	4		600.2.k.c.301.3		6
80.67	even	4		600.2.d.e.349.6		6
80.69	even	4		2400.2.k.c.1201.5		6
80.77	odd	4		2400.2.d.f.49.3		6
240.29	odd	4		7200.2.k.p.3601.3		6
240.53	even	4		7200.2.d.q.2449.4		6
240.59	even	4		1800.2.k.p.901.4		6
240.77	even	4		7200.2.d.q.2449.3		6
240.83	odd	4		1800.2.d.r.1549.6		6
240.107	odd	4		1800.2.d.r.1549.5		6
240.149	odd	4		7200.2.k.p.3601.4		6
240.173	even	4		7200.2.d.r.2449.4		6
240.179	even	4		1800.2.k.p.901.3		6
240.197	even	4		7200.2.d.r.2449.3		6
240.203	odd	4		1800.2.d.q.1549.2		6
240.227	odd	4		1800.2.d.q.1549.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.k.b.61.3	✓	6	16.3	odd	4
120.2.k.b.61.4	yes	6	16.11	odd	4
360.2.k.f.181.3		6	48.11	even	4
360.2.k.f.181.4		6	48.35	even	4
480.2.k.b.241.2		6	16.5	even	4
480.2.k.b.241.5		6	16.13	even	4
600.2.d.e.349.5		6	80.43	even	4
600.2.d.e.349.6		6	80.67	even	4
600.2.d.f.349.1		6	80.3	even	4
600.2.d.f.349.2		6	80.27	even	4
600.2.k.c.301.3		6	80.59	odd	4
600.2.k.c.301.4		6	80.19	odd	4
1440.2.k.f.721.2		6	48.5	odd	4
1440.2.k.f.721.5		6	48.29	odd	4
1800.2.d.q.1549.1		6	240.227	odd	4
1800.2.d.q.1549.2		6	240.203	odd	4
1800.2.d.r.1549.5		6	240.107	odd	4
1800.2.d.r.1549.6		6	240.83	odd	4
1800.2.k.p.901.3		6	240.179	even	4
1800.2.k.p.901.4		6	240.59	even	4
2400.2.d.e.49.3		6	80.37	odd	4
2400.2.d.e.49.4		6	80.13	odd	4
2400.2.d.f.49.3		6	80.77	odd	4
2400.2.d.f.49.4		6	80.53	odd	4
2400.2.k.c.1201.2		6	80.29	even	4
2400.2.k.c.1201.5		6	80.69	even	4
3840.2.a.bo.1.2		3	8.5	even	2
3840.2.a.bp.1.2		3	4.3	odd	2
3840.2.a.bq.1.2		3	8.3	odd	2
3840.2.a.br.1.2		3	1.1	even	1	trivial
7200.2.d.q.2449.3		6	240.77	even	4
7200.2.d.q.2449.4		6	240.53	even	4
7200.2.d.r.2449.3		6	240.197	even	4
7200.2.d.r.2449.4		6	240.173	even	4
7200.2.k.p.3601.3		6	240.29	odd	4
7200.2.k.p.3601.4		6	240.149	odd	4

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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +0.941367 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +0.941367 q^{7} +1.00000 q^{9} -4.49828 q^{11} +5.55691 q^{13} +1.00000 q^{15} +7.55691 q^{17} +1.05863 q^{19} +0.941367 q^{21} -1.05863 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} -3.55691 q^{31} -4.49828 q^{33} +0.941367 q^{35} -7.43965 q^{37} +5.55691 q^{39} +3.88273 q^{41} +1.88273 q^{43} +1.00000 q^{45} +10.0552 q^{47} -6.11383 q^{49} +7.55691 q^{51} +2.00000 q^{53} -4.49828 q^{55} +1.05863 q^{57} +8.49828 q^{59} -8.99656 q^{61} +0.941367 q^{63} +5.55691 q^{65} -4.00000 q^{67} -1.05863 q^{69} -12.9966 q^{71} +6.00000 q^{73} +1.00000 q^{75} -4.23453 q^{77} -11.5569 q^{79} +1.00000 q^{81} +5.88273 q^{83} +7.55691 q^{85} +2.00000 q^{87} +4.11727 q^{89} +5.23109 q^{91} -3.55691 q^{93} +1.05863 q^{95} +17.1138 q^{97} -4.49828 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9} + 4 q^{11} + 3 q^{15} + 6 q^{17} + 4 q^{19} + 2 q^{21} - 4 q^{23} + 3 q^{25} + 3 q^{27} + 6 q^{29} + 6 q^{31} + 4 q^{33} + 2 q^{35} - 4 q^{37} + 10 q^{41} + 4 q^{43} + 3 q^{45} - 4 q^{47} + 15 q^{49} + 6 q^{51} + 6 q^{53} + 4 q^{55} + 4 q^{57} + 8 q^{59} + 8 q^{61} + 2 q^{63} - 12 q^{67} - 4 q^{69} - 4 q^{71} + 18 q^{73} + 3 q^{75} - 16 q^{77} - 18 q^{79} + 3 q^{81} + 16 q^{83} + 6 q^{85} + 6 q^{87} + 14 q^{89} - 16 q^{91} + 6 q^{93} + 4 q^{95} + 18 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 3 * q^9 + 4 * q^11 + 3 * q^15 + 6 * q^17 + 4 * q^19 + 2 * q^21 - 4 * q^23 + 3 * q^25 + 3 * q^27 + 6 * q^29 + 6 * q^31 + 4 * q^33 + 2 * q^35 - 4 * q^37 + 10 * q^41 + 4 * q^43 + 3 * q^45 - 4 * q^47 + 15 * q^49 + 6 * q^51 + 6 * q^53 + 4 * q^55 + 4 * q^57 + 8 * q^59 + 8 * q^61 + 2 * q^63 - 12 * q^67 - 4 * q^69 - 4 * q^71 + 18 * q^73 + 3 * q^75 - 16 * q^77 - 18 * q^79 + 3 * q^81 + 16 * q^83 + 6 * q^85 + 6 * q^87 + 14 * q^89 - 16 * q^91 + 6 * q^93 + 4 * q^95 + 18 * q^97 + 4 * q^99

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