LMFDB - Embedded newform 4732.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4732,2,Mod(1,4732)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4732, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4732.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4732 = 2^{2} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4732.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:	$37.7852102365$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.25492.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} - 2x + 8 $ x^4 - 8x^2 - 2x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 364)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.29363$ of defining polynomial
Character	$\chi$	$=$	4732.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.29363 q^{3} +2.79846 q^{5} +1.00000 q^{7} -1.32653 q^{9} +O(q^{10})$	$q-1.29363 q^{3} +2.79846 q^{5} +1.00000 q^{7} -1.32653 q^{9} +1.03291 q^{11} -3.62016 q^{15} +6.95635 q^{17} +2.53774 q^{19} -1.29363 q^{21} +3.83136 q^{23} +2.83136 q^{25} +5.59691 q^{27} -5.10175 q^{29} +3.78880 q^{31} -1.33619 q^{33} +2.79846 q^{35} -6.20741 q^{37} -0.990338 q^{41} -0.560979 q^{43} -3.71224 q^{45} -7.19080 q^{47} +1.00000 q^{49} -8.99892 q^{51} +1.17830 q^{53} +2.89054 q^{55} -3.28288 q^{57} +11.9234 q^{59} +13.8469 q^{61} -1.32653 q^{63} -13.7811 q^{67} -4.95635 q^{69} -3.80432 q^{71} +3.19080 q^{73} -3.66272 q^{75} +1.03291 q^{77} +7.33240 q^{79} -3.26072 q^{81} +16.6260 q^{83} +19.4670 q^{85} +6.59975 q^{87} -11.8964 q^{89} -4.90128 q^{93} +7.10175 q^{95} +8.60355 q^{97} -1.37018 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 3 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q + 3 q^{5} + 4 q^{7} + 4 q^{9} - 2 q^{17} + 3 q^{19} + 3 q^{23} - q^{25} + 6 q^{27} - q^{29} + 13 q^{31} + 10 q^{33} + 3 q^{35} - 10 q^{41} + 3 q^{43} + 13 q^{45} - 3 q^{47} + 4 q^{49} + 4 q^{51} + 11 q^{53} - 10 q^{55} + 26 q^{57} + 22 q^{59} + 4 q^{61} + 4 q^{63} - 12 q^{67} + 10 q^{69} + 26 q^{71} - 13 q^{73} + 10 q^{75} - 13 q^{79} - 12 q^{81} + 19 q^{83} + 12 q^{85} - 16 q^{87} + 7 q^{89} - 32 q^{93} + 9 q^{95} + 19 q^{97} - 26 q^{99}+O(q^{100}) $ 4 * q + 3 * q^5 + 4 * q^7 + 4 * q^9 - 2 * q^17 + 3 * q^19 + 3 * q^23 - q^25 + 6 * q^27 - q^29 + 13 * q^31 + 10 * q^33 + 3 * q^35 - 10 * q^41 + 3 * q^43 + 13 * q^45 - 3 * q^47 + 4 * q^49 + 4 * q^51 + 11 * q^53 - 10 * q^55 + 26 * q^57 + 22 * q^59 + 4 * q^61 + 4 * q^63 - 12 * q^67 + 10 * q^69 + 26 * q^71 - 13 * q^73 + 10 * q^75 - 13 * q^79 - 12 * q^81 + 19 * q^83 + 12 * q^85 - 16 * q^87 + 7 * q^89 - 32 * q^93 + 9 * q^95 + 19 * q^97 - 26 * 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
$3$	−1.29363	−0.746875	−0.373438	−	0.927655i	$-0.621821\pi$
$3$	−1.29363	−0.746875	−0.373438	+	0.927655i	$0.621821\pi$
$4$	0	0
$5$	2.79846	1.25151	0.625754	−	0.780020i	$-0.284791\pi$
$5$	2.79846	1.25151	0.625754	+	0.780020i	$0.284791\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.32653	−0.442177
$10$	0	0
$11$	1.03291	0.311433	0.155716	−	0.987802i	$-0.450231\pi$
$11$	1.03291	0.311433	0.155716	+	0.987802i	$0.450231\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−3.62016	−0.934721
$16$	0	0
$17$	6.95635	1.68716	0.843581	−	0.537001i	$-0.180443\pi$
$17$	6.95635	1.68716	0.843581	+	0.537001i	$0.180443\pi$
$18$	0	0
$19$	2.53774	0.582197	0.291098	−	0.956693i	$-0.405979\pi$
$19$	2.53774	0.582197	0.291098	+	0.956693i	$0.405979\pi$
$20$	0	0
$21$	−1.29363	−0.282292
$22$	0	0
$23$	3.83136	0.798894	0.399447	−	0.916756i	$-0.369202\pi$
$23$	3.83136	0.798894	0.399447	+	0.916756i	$0.369202\pi$
$24$	0	0
$25$	2.83136	0.566272
$26$	0	0
$27$	5.59691	1.07713
$28$	0	0
$29$	−5.10175	−0.947370	−0.473685	−	0.880694i	$-0.657077\pi$
$29$	−5.10175	−0.947370	−0.473685	+	0.880694i	$0.657077\pi$
$30$	0	0
$31$	3.78880	0.680488	0.340244	−	0.940337i	$-0.389490\pi$
$31$	3.78880	0.680488	0.340244	+	0.940337i	$0.389490\pi$
$32$	0	0
$33$	−1.33619	−0.232601
$34$	0	0
$35$	2.79846	0.473026
$36$	0	0
$37$	−6.20741	−1.02049	−0.510246	−	0.860029i	$-0.670446\pi$
$37$	−6.20741	−1.02049	−0.510246	+	0.860029i	$0.670446\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.990338	−0.154665	−0.0773324	−	0.997005i	$-0.524640\pi$
$41$	−0.990338	−0.154665	−0.0773324	+	0.997005i	$0.524640\pi$
$42$	0	0
$43$	−0.560979	−0.0855485	−0.0427743	−	0.999085i	$-0.513620\pi$
$43$	−0.560979	−0.0855485	−0.0427743	+	0.999085i	$0.513620\pi$
$44$	0	0
$45$	−3.71224	−0.553388
$46$	0	0
$47$	−7.19080	−1.04889	−0.524443	−	0.851446i	$-0.675726\pi$
$47$	−7.19080	−1.04889	−0.524443	+	0.851446i	$0.675726\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−8.99892	−1.26010
$52$	0	0
$53$	1.17830	0.161852	0.0809260	−	0.996720i	$-0.474212\pi$
$53$	1.17830	0.161852	0.0809260	+	0.996720i	$0.474212\pi$
$54$	0	0
$55$	2.89054	0.389760
$56$	0	0
$57$	−3.28288	−0.434828
$58$	0	0
$59$	11.9234	1.55230	0.776150	−	0.630548i	$-0.217170\pi$
$59$	11.9234	1.55230	0.776150	+	0.630548i	$0.217170\pi$
$60$	0	0
$61$	13.8469	1.77291	0.886456	−	0.462812i	$-0.153160\pi$
$61$	13.8469	1.77291	0.886456	+	0.462812i	$0.153160\pi$
$62$	0	0
$63$	−1.32653	−0.167127
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.7811	−1.68363	−0.841813	−	0.539769i	$-0.818512\pi$
$67$	−13.7811	−1.68363	−0.841813	+	0.539769i	$0.818512\pi$
$68$	0	0
$69$	−4.95635	−0.596674
$70$	0	0
$71$	−3.80432	−0.451490	−0.225745	−	0.974186i	$-0.572482\pi$
$71$	−3.80432	−0.451490	−0.225745	+	0.974186i	$0.572482\pi$
$72$	0	0
$73$	3.19080	0.373455	0.186727	−	0.982412i	$-0.440212\pi$
$73$	3.19080	0.373455	0.186727	+	0.982412i	$0.440212\pi$
$74$	0	0
$75$	−3.66272	−0.422935
$76$	0	0
$77$	1.03291	0.117710
$78$	0	0
$79$	7.33240	0.824959	0.412480	−	0.910967i	$-0.364663\pi$
$79$	7.33240	0.824959	0.412480	+	0.910967i	$0.364663\pi$
$80$	0	0
$81$	−3.26072	−0.362302
$82$	0	0
$83$	16.6260	1.82494	0.912472	−	0.409140i	$-0.134171\pi$
$83$	16.6260	1.82494	0.912472	+	0.409140i	$0.134171\pi$
$84$	0	0
$85$	19.4670	2.11150
$86$	0	0
$87$	6.59975	0.707568
$88$	0	0
$89$	−11.8964	−1.26102	−0.630508	−	0.776182i	$-0.717154\pi$
$89$	−11.8964	−1.26102	−0.630508	+	0.776182i	$0.717154\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.90128	−0.508240
$94$	0	0
$95$	7.10175	0.728624
$96$	0	0
$97$	8.60355	0.873558	0.436779	−	0.899569i	$-0.356119\pi$
$97$	8.60355	0.873558	0.436779	+	0.899569i	$0.356119\pi$
$98$	0	0
$99$	−1.37018	−0.137708
$100$	0	0
$101$	−8.36910	−0.832756	−0.416378	−	0.909192i	$-0.636701\pi$
$101$	−8.36910	−0.832756	−0.416378	+	0.909192i	$0.636701\pi$
$102$	0	0
$103$	−0.521442	−0.0513792	−0.0256896	−	0.999670i	$-0.508178\pi$
$103$	−0.521442	−0.0513792	−0.0256896	+	0.999670i	$0.508178\pi$
$104$	0	0
$105$	−3.62016	−0.353291
$106$	0	0
$107$	9.43523	0.912138	0.456069	−	0.889944i	$-0.349257\pi$
$107$	9.43523	0.912138	0.456069	+	0.889944i	$0.349257\pi$
$108$	0	0
$109$	13.6860	1.31088	0.655439	−	0.755248i	$-0.272484\pi$
$109$	13.6860	1.31088	0.655439	+	0.755248i	$0.272484\pi$
$110$	0	0
$111$	8.03007	0.762180
$112$	0	0
$113$	−12.2655	−1.15384	−0.576921	−	0.816800i	$-0.695746\pi$
$113$	−12.2655	−1.15384	−0.576921	+	0.816800i	$0.695746\pi$
$114$	0	0
$115$	10.7219	0.999823
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.95635	0.637688
$120$	0	0
$121$	−9.93311	−0.903010
$122$	0	0
$123$	1.28113	0.115515
$124$	0	0
$125$	−6.06884	−0.542814
$126$	0	0
$127$	10.9331	0.970156	0.485078	−	0.874471i	$-0.338791\pi$
$127$	10.9331	0.970156	0.485078	+	0.874471i	$0.338791\pi$
$128$	0	0
$129$	0.725697	0.0638941
$130$	0	0
$131$	−6.58725	−0.575531	−0.287765	−	0.957701i	$-0.592912\pi$
$131$	−6.58725	−0.575531	−0.287765	+	0.957701i	$0.592912\pi$
$132$	0	0
$133$	2.53774	0.220050
$134$	0	0
$135$	15.6627	1.34803
$136$	0	0
$137$	−12.8333	−1.09642	−0.548212	−	0.836340i	$-0.684691\pi$
$137$	−12.8333	−1.09642	−0.548212	+	0.836340i	$0.684691\pi$
$138$	0	0
$139$	−18.5998	−1.57761	−0.788805	−	0.614643i	$-0.789300\pi$
$139$	−18.5998	−1.57761	−0.788805	+	0.614643i	$0.789300\pi$
$140$	0	0
$141$	9.30221	0.783387
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−14.2770	−1.18564
$146$	0	0
$147$	−1.29363	−0.106696
$148$	0	0
$149$	9.97676	0.817328	0.408664	−	0.912685i	$-0.365995\pi$
$149$	9.97676	0.817328	0.408664	+	0.912685i	$0.365995\pi$
$150$	0	0
$151$	14.0543	1.14372	0.571861	−	0.820350i	$-0.306222\pi$
$151$	14.0543	1.14372	0.571861	+	0.820350i	$0.306222\pi$
$152$	0	0
$153$	−9.22782	−0.746025
$154$	0	0
$155$	10.6028	0.851636
$156$	0	0
$157$	−7.76858	−0.620000	−0.310000	−	0.950737i	$-0.600329\pi$
$157$	−7.76858	−0.620000	−0.310000	+	0.950737i	$0.600329\pi$
$158$	0	0
$159$	−1.52428	−0.120883
$160$	0	0
$161$	3.83136	0.301954
$162$	0	0
$163$	7.39342	0.579098	0.289549	−	0.957163i	$-0.406495\pi$
$163$	7.39342	0.579098	0.289549	+	0.957163i	$0.406495\pi$
$164$	0	0
$165$	−3.73928	−0.291102
$166$	0	0
$167$	15.7122	1.21585	0.607925	−	0.793995i	$-0.292002\pi$
$167$	15.7122	1.21585	0.607925	+	0.793995i	$0.292002\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−3.36639	−0.257434
$172$	0	0
$173$	0.415585	0.0315963	0.0157982	−	0.999875i	$-0.494971\pi$
$173$	0.415585	0.0315963	0.0157982	+	0.999875i	$0.494971\pi$
$174$	0	0
$175$	2.83136	0.214031
$176$	0	0
$177$	−15.4245	−1.15938
$178$	0	0
$179$	−3.90792	−0.292091	−0.146046	−	0.989278i	$-0.546655\pi$
$179$	−3.90792	−0.292091	−0.146046	+	0.989278i	$0.546655\pi$
$180$	0	0
$181$	−4.79151	−0.356150	−0.178075	−	0.984017i	$-0.556987\pi$
$181$	−4.79151	−0.356150	−0.178075	+	0.984017i	$0.556987\pi$
$182$	0	0
$183$	−17.9127	−1.32414
$184$	0	0
$185$	−17.3712	−1.27715
$186$	0	0
$187$	7.18525	0.525437
$188$	0	0
$189$	5.59691	0.407116
$190$	0	0
$191$	−8.62905	−0.624376	−0.312188	−	0.950020i	$-0.601062\pi$
$191$	−8.62905	−0.624376	−0.312188	+	0.950020i	$0.601062\pi$
$192$	0	0
$193$	−24.4535	−1.76020	−0.880100	−	0.474789i	$-0.842524\pi$
$193$	−24.4535	−1.76020	−0.880100	+	0.474789i	$0.842524\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.14128	−0.508795	−0.254398	−	0.967100i	$-0.581877\pi$
$197$	−7.14128	−0.508795	−0.254398	+	0.967100i	$0.581877\pi$
$198$	0	0
$199$	1.00284	0.0710892	0.0355446	−	0.999368i	$-0.488683\pi$
$199$	1.00284	0.0710892	0.0355446	+	0.999368i	$0.488683\pi$
$200$	0	0
$201$	17.8276	1.25746
$202$	0	0
$203$	−5.10175	−0.358072
$204$	0	0
$205$	−2.77142	−0.193564
$206$	0	0
$207$	−5.08242	−0.353253
$208$	0	0
$209$	2.62124	0.181315
$210$	0	0
$211$	26.2655	1.80819	0.904096	−	0.427329i	$-0.140546\pi$
$211$	26.2655	1.80819	0.904096	+	0.427329i	$0.140546\pi$
$212$	0	0
$213$	4.92137	0.337207
$214$	0	0
$215$	−1.56988	−0.107065
$216$	0	0
$217$	3.78880	0.257200
$218$	0	0
$219$	−4.12770	−0.278924
$220$	0	0
$221$	0	0
$222$	0	0
$223$	17.7540	1.18890	0.594449	−	0.804133i	$-0.297370\pi$
$223$	17.7540	1.18890	0.594449	+	0.804133i	$0.297370\pi$
$224$	0	0
$225$	−3.75589	−0.250393
$226$	0	0
$227$	17.8276	1.18326	0.591629	−	0.806211i	$-0.298485\pi$
$227$	17.8276	1.18326	0.591629	+	0.806211i	$0.298485\pi$
$228$	0	0
$229$	4.21423	0.278484	0.139242	−	0.990258i	$-0.455533\pi$
$229$	4.21423	0.278484	0.139242	+	0.990258i	$0.455533\pi$
$230$	0	0
$231$	−1.33619	−0.0879150
$232$	0	0
$233$	28.2880	1.85321	0.926604	−	0.376039i	$-0.122714\pi$
$233$	28.2880	1.85321	0.926604	+	0.376039i	$0.122714\pi$
$234$	0	0
$235$	−20.1231	−1.31269
$236$	0	0
$237$	−9.48538	−0.616142
$238$	0	0
$239$	8.94385	0.578530	0.289265	−	0.957249i	$-0.406589\pi$
$239$	8.94385	0.578530	0.289265	+	0.957249i	$0.406589\pi$
$240$	0	0
$241$	17.0019	1.09519	0.547596	−	0.836743i	$-0.315543\pi$
$241$	17.0019	1.09519	0.547596	+	0.836743i	$0.315543\pi$
$242$	0	0
$243$	−12.5726	−0.806532
$244$	0	0
$245$	2.79846	0.178787
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−21.5079	−1.36301
$250$	0	0
$251$	6.70637	0.423303	0.211651	−	0.977345i	$-0.432116\pi$
$251$	6.70637	0.423303	0.211651	+	0.977345i	$0.432116\pi$
$252$	0	0
$253$	3.95743	0.248802
$254$	0	0
$255$	−25.1831	−1.57703
$256$	0	0
$257$	14.6656	0.914813	0.457406	−	0.889258i	$-0.348779\pi$
$257$	14.6656	0.914813	0.457406	+	0.889258i	$0.348779\pi$
$258$	0	0
$259$	−6.20741	−0.385710
$260$	0	0
$261$	6.76762	0.418905
$262$	0	0
$263$	−6.02627	−0.371596	−0.185798	−	0.982588i	$-0.559487\pi$
$263$	−6.02627	−0.371596	−0.185798	+	0.982588i	$0.559487\pi$
$264$	0	0
$265$	3.29742	0.202559
$266$	0	0
$267$	15.3895	0.941822
$268$	0	0
$269$	−0.825495	−0.0503313	−0.0251657	−	0.999683i	$-0.508011\pi$
$269$	−0.825495	−0.0503313	−0.0251657	+	0.999683i	$0.508011\pi$
$270$	0	0
$271$	9.82757	0.596982	0.298491	−	0.954412i	$-0.403517\pi$
$271$	9.82757	0.596982	0.298491	+	0.954412i	$0.403517\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.92453	0.176356
$276$	0	0
$277$	−27.8517	−1.67344	−0.836722	−	0.547627i	$-0.815531\pi$
$277$	−27.8517	−1.67344	−0.836722	+	0.547627i	$0.815531\pi$
$278$	0	0
$279$	−5.02596	−0.300896
$280$	0	0
$281$	−4.04257	−0.241159	−0.120580	−	0.992704i	$-0.538475\pi$
$281$	−4.04257	−0.241159	−0.120580	+	0.992704i	$0.538475\pi$
$282$	0	0
$283$	29.7596	1.76902	0.884512	−	0.466517i	$-0.154491\pi$
$283$	29.7596	1.76902	0.884512	+	0.466517i	$0.154491\pi$
$284$	0	0
$285$	−9.18700	−0.544191
$286$	0	0
$287$	−0.990338	−0.0584578
$288$	0	0
$289$	31.3908	1.84652
$290$	0	0
$291$	−11.1298	−0.652439
$292$	0	0
$293$	29.8091	1.74147	0.870733	−	0.491755i	$-0.163645\pi$
$293$	29.8091	1.74147	0.870733	+	0.491755i	$0.163645\pi$
$294$	0	0
$295$	33.3673	1.94272
$296$	0	0
$297$	5.78108	0.335452
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−0.560979	−0.0323343
$302$	0	0
$303$	10.8265	0.621965
$304$	0	0
$305$	38.7499	2.21881
$306$	0	0
$307$	−18.5928	−1.06115	−0.530574	−	0.847639i	$-0.678023\pi$
$307$	−18.5928	−1.06115	−0.530574	+	0.847639i	$0.678023\pi$
$308$	0	0
$309$	0.674552	0.0383739
$310$	0	0
$311$	−15.9119	−0.902283	−0.451142	−	0.892452i	$-0.648983\pi$
$311$	−15.9119	−0.902283	−0.451142	+	0.892452i	$0.648983\pi$
$312$	0	0
$313$	−1.79575	−0.101502	−0.0507508	−	0.998711i	$-0.516161\pi$
$313$	−1.79575	−0.101502	−0.0507508	+	0.998711i	$0.516161\pi$
$314$	0	0
$315$	−3.71224	−0.209161
$316$	0	0
$317$	−16.0311	−0.900394	−0.450197	−	0.892929i	$-0.648646\pi$
$317$	−16.0311	−0.900394	−0.450197	+	0.892929i	$0.648646\pi$
$318$	0	0
$319$	−5.26962	−0.295042
$320$	0	0
$321$	−12.2057	−0.681253
$322$	0	0
$323$	17.6534	0.982260
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−17.7045	−0.979063
$328$	0	0
$329$	−7.19080	−0.396442
$330$	0	0
$331$	6.81791	0.374746	0.187373	−	0.982289i	$-0.440003\pi$
$331$	6.81791	0.374746	0.187373	+	0.982289i	$0.440003\pi$
$332$	0	0
$333$	8.23432	0.451238
$334$	0	0
$335$	−38.5658	−2.10707
$336$	0	0
$337$	−0.418615	−0.0228034	−0.0114017	−	0.999935i	$-0.503629\pi$
$337$	−0.418615	−0.0228034	−0.0114017	+	0.999935i	$0.503629\pi$
$338$	0	0
$339$	15.8670	0.861776
$340$	0	0
$341$	3.91347	0.211926
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−13.8701	−0.746743
$346$	0	0
$347$	−10.0658	−0.540361	−0.270180	−	0.962810i	$-0.587083\pi$
$347$	−10.0658	−0.540361	−0.270180	+	0.962810i	$0.587083\pi$
$348$	0	0
$349$	8.27233	0.442808	0.221404	−	0.975182i	$-0.428936\pi$
$349$	8.27233	0.442808	0.221404	+	0.975182i	$0.428936\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	16.3908	0.872395	0.436197	−	0.899851i	$-0.356325\pi$
$353$	16.3908	0.872395	0.436197	+	0.899851i	$0.356325\pi$
$354$	0	0
$355$	−10.6462	−0.565044
$356$	0	0
$357$	−8.99892	−0.476273
$358$	0	0
$359$	32.1512	1.69687	0.848437	−	0.529297i	$-0.177544\pi$
$359$	32.1512	1.69687	0.848437	+	0.529297i	$0.177544\pi$
$360$	0	0
$361$	−12.5599	−0.661047
$362$	0	0
$363$	12.8497	0.674436
$364$	0	0
$365$	8.92931	0.467382
$366$	0	0
$367$	−7.79575	−0.406935	−0.203467	−	0.979082i	$-0.565221\pi$
$367$	−7.79575	−0.406935	−0.203467	+	0.979082i	$0.565221\pi$
$368$	0	0
$369$	1.31371	0.0683892
$370$	0	0
$371$	1.17830	0.0611743
$372$	0	0
$373$	31.7371	1.64329	0.821643	−	0.570003i	$-0.193058\pi$
$373$	31.7371	1.64329	0.821643	+	0.570003i	$0.193058\pi$
$374$	0	0
$375$	7.85081	0.405414
$376$	0	0
$377$	0	0
$378$	0	0
$379$	30.3236	1.55762	0.778809	−	0.627261i	$-0.215824\pi$
$379$	30.3236	1.55762	0.778809	+	0.627261i	$0.215824\pi$
$380$	0	0
$381$	−14.1434	−0.724586
$382$	0	0
$383$	14.0193	0.716354	0.358177	−	0.933654i	$-0.383398\pi$
$383$	14.0193	0.716354	0.358177	+	0.933654i	$0.383398\pi$
$384$	0	0
$385$	2.89054	0.147316
$386$	0	0
$387$	0.744156	0.0378276
$388$	0	0
$389$	−19.8887	−1.00840	−0.504198	−	0.863588i	$-0.668212\pi$
$389$	−19.8887	−1.00840	−0.504198	+	0.863588i	$0.668212\pi$
$390$	0	0
$391$	26.6523	1.34786
$392$	0	0
$393$	8.52144	0.429850
$394$	0	0
$395$	20.5194	1.03244
$396$	0	0
$397$	−14.7219	−0.738871	−0.369436	−	0.929256i	$-0.620449\pi$
$397$	−14.7219	−0.738871	−0.369436	+	0.929256i	$0.620449\pi$
$398$	0	0
$399$	−3.28288	−0.164350
$400$	0	0
$401$	19.6412	0.980836	0.490418	−	0.871487i	$-0.336844\pi$
$401$	19.6412	0.980836	0.490418	+	0.871487i	$0.336844\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−9.12499	−0.453424
$406$	0	0
$407$	−6.41166	−0.317814
$408$	0	0
$409$	−24.1132	−1.19232	−0.596160	−	0.802866i	$-0.703307\pi$
$409$	−24.1132	−1.19232	−0.596160	+	0.802866i	$0.703307\pi$
$410$	0	0
$411$	16.6015	0.818892
$412$	0	0
$413$	11.9234	0.586714
$414$	0	0
$415$	46.5272	2.28393
$416$	0	0
$417$	24.0611	1.17828
$418$	0	0
$419$	−19.7818	−0.966406	−0.483203	−	0.875508i	$-0.660527\pi$
$419$	−19.7818	−0.966406	−0.483203	+	0.875508i	$0.660527\pi$
$420$	0	0
$421$	19.1237	0.932033	0.466016	−	0.884776i	$-0.345689\pi$
$421$	19.1237	0.932033	0.466016	+	0.884776i	$0.345689\pi$
$422$	0	0
$423$	9.53882	0.463793
$424$	0	0
$425$	19.6959	0.955394
$426$	0	0
$427$	13.8469	0.670098
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.87588	−0.427536	−0.213768	−	0.976884i	$-0.568574\pi$
$431$	−8.87588	−0.427536	−0.213768	+	0.976884i	$0.568574\pi$
$432$	0	0
$433$	36.9341	1.77494	0.887470	−	0.460866i	$-0.152461\pi$
$433$	36.9341	1.77494	0.887470	+	0.460866i	$0.152461\pi$
$434$	0	0
$435$	18.4691	0.885527
$436$	0	0
$437$	9.72299	0.465113
$438$	0	0
$439$	20.6591	0.986006	0.493003	−	0.870028i	$-0.335899\pi$
$439$	20.6591	0.986006	0.493003	+	0.870028i	$0.335899\pi$
$440$	0	0
$441$	−1.32653	−0.0631682
$442$	0	0
$443$	2.85492	0.135641	0.0678207	−	0.997698i	$-0.478395\pi$
$443$	2.85492	0.135641	0.0678207	+	0.997698i	$0.478395\pi$
$444$	0	0
$445$	−33.2916	−1.57817
$446$	0	0
$447$	−12.9062	−0.610442
$448$	0	0
$449$	−30.0443	−1.41788	−0.708940	−	0.705269i	$-0.750826\pi$
$449$	−30.0443	−1.41788	−0.708940	+	0.705269i	$0.750826\pi$
$450$	0	0
$451$	−1.02293	−0.0481677
$452$	0	0
$453$	−18.1810	−0.854218
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.6491	−1.34015	−0.670075	−	0.742293i	$-0.733738\pi$
$457$	−28.6491	−1.34015	−0.670075	+	0.742293i	$0.733738\pi$
$458$	0	0
$459$	38.9341	1.81729
$460$	0	0
$461$	−0.922363	−0.0429587	−0.0214794	−	0.999769i	$-0.506838\pi$
$461$	−0.922363	−0.0429587	−0.0214794	+	0.999769i	$0.506838\pi$
$462$	0	0
$463$	−26.0929	−1.21264	−0.606321	−	0.795220i	$-0.707355\pi$
$463$	−26.0929	−1.21264	−0.606321	+	0.795220i	$0.707355\pi$
$464$	0	0
$465$	−13.7160	−0.636066
$466$	0	0
$467$	−29.5553	−1.36766	−0.683829	−	0.729642i	$-0.739687\pi$
$467$	−29.5553	−1.36766	−0.683829	+	0.729642i	$0.739687\pi$
$468$	0	0
$469$	−13.7811	−0.636351
$470$	0	0
$471$	10.0496	0.463063
$472$	0	0
$473$	−0.579438	−0.0266426
$474$	0	0
$475$	7.18525	0.329682
$476$	0	0
$477$	−1.56305	−0.0715672
$478$	0	0
$479$	15.8954	0.726280	0.363140	−	0.931735i	$-0.381705\pi$
$479$	15.8954	0.726280	0.363140	+	0.931735i	$0.381705\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−4.95635	−0.225522
$484$	0	0
$485$	24.0767	1.09326
$486$	0	0
$487$	0.0851336	0.00385777	0.00192889	−	0.999998i	$-0.499386\pi$
$487$	0.0851336	0.00385777	0.00192889	+	0.999998i	$0.499386\pi$
$488$	0	0
$489$	−9.56433	−0.432514
$490$	0	0
$491$	−2.56324	−0.115678	−0.0578388	−	0.998326i	$-0.518421\pi$
$491$	−2.56324	−0.115678	−0.0578388	+	0.998326i	$0.518421\pi$
$492$	0	0
$493$	−35.4895	−1.59837
$494$	0	0
$495$	−3.83439	−0.172343
$496$	0	0
$497$	−3.80432	−0.170647
$498$	0	0
$499$	28.6941	1.28452	0.642262	−	0.766485i	$-0.277996\pi$
$499$	28.6941	1.28452	0.642262	+	0.766485i	$0.277996\pi$
$500$	0	0
$501$	−20.3258	−0.908088
$502$	0	0
$503$	−32.7056	−1.45827	−0.729136	−	0.684369i	$-0.760078\pi$
$503$	−32.7056	−1.45827	−0.729136	+	0.684369i	$0.760078\pi$
$504$	0	0
$505$	−23.4206	−1.04220
$506$	0	0
$507$	0	0
$508$	0	0
$509$	26.0938	1.15659	0.578295	−	0.815828i	$-0.303718\pi$
$509$	26.0938	1.15659	0.578295	+	0.815828i	$0.303718\pi$
$510$	0	0
$511$	3.19080	0.141153
$512$	0	0
$513$	14.2035	0.627099
$514$	0	0
$515$	−1.45923	−0.0643015
$516$	0	0
$517$	−7.42741	−0.326657
$518$	0	0
$519$	−0.537611	−0.0235985
$520$	0	0
$521$	−1.75570	−0.0769185	−0.0384592	−	0.999260i	$-0.512245\pi$
$521$	−1.75570	−0.0769185	−0.0384592	+	0.999260i	$0.512245\pi$
$522$	0	0
$523$	−32.3779	−1.41579	−0.707893	−	0.706319i	$-0.750354\pi$
$523$	−32.3779	−1.41579	−0.707893	+	0.706319i	$0.750354\pi$
$524$	0	0
$525$	−3.66272	−0.159854
$526$	0	0
$527$	26.3562	1.14809
$528$	0	0
$529$	−8.32066	−0.361768
$530$	0	0
$531$	−15.8168	−0.686392
$532$	0	0
$533$	0	0
$534$	0	0
$535$	26.4041	1.14155
$536$	0	0
$537$	5.05538	0.218156
$538$	0	0
$539$	1.03291	0.0444904
$540$	0	0
$541$	8.52536	0.366534	0.183267	−	0.983063i	$-0.441333\pi$
$541$	8.52536	0.366534	0.183267	+	0.983063i	$0.441333\pi$
$542$	0	0
$543$	6.19842	0.266000
$544$	0	0
$545$	38.2996	1.64057
$546$	0	0
$547$	1.14355	0.0488945	0.0244473	−	0.999701i	$-0.492217\pi$
$547$	1.14355	0.0488945	0.0244473	+	0.999701i	$0.492217\pi$
$548$	0	0
$549$	−18.3683	−0.783941
$550$	0	0
$551$	−12.9469	−0.551556
$552$	0	0
$553$	7.33240	0.311805
$554$	0	0
$555$	22.4718	0.953875
$556$	0	0
$557$	−42.8275	−1.81466	−0.907329	−	0.420421i	$-0.861882\pi$
$557$	−42.8275	−1.81466	−0.907329	+	0.420421i	$0.861882\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−9.29503	−0.392436
$562$	0	0
$563$	−13.5754	−0.572136	−0.286068	−	0.958209i	$-0.592348\pi$
$563$	−13.5754	−0.572136	−0.286068	+	0.958209i	$0.592348\pi$
$564$	0	0
$565$	−34.3245	−1.44404
$566$	0	0
$567$	−3.26072	−0.136937
$568$	0	0
$569$	8.87109	0.371896	0.185948	−	0.982560i	$-0.440464\pi$
$569$	8.87109	0.371896	0.185948	+	0.982560i	$0.440464\pi$
$570$	0	0
$571$	−6.28093	−0.262849	−0.131424	−	0.991326i	$-0.541955\pi$
$571$	−6.28093	−0.262849	−0.131424	+	0.991326i	$0.541955\pi$
$572$	0	0
$573$	11.1628	0.466331
$574$	0	0
$575$	10.8480	0.452392
$576$	0	0
$577$	−25.1005	−1.04495	−0.522473	−	0.852656i	$-0.674990\pi$
$577$	−25.1005	−1.04495	−0.522473	+	0.852656i	$0.674990\pi$
$578$	0	0
$579$	31.6336	1.31465
$580$	0	0
$581$	16.6260	0.689764
$582$	0	0
$583$	1.21707	0.0504060
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.2963	1.16791	0.583957	−	0.811784i	$-0.301503\pi$
$587$	28.2963	1.16791	0.583957	+	0.811784i	$0.301503\pi$
$588$	0	0
$589$	9.61496	0.396178
$590$	0	0
$591$	9.23815	0.380007
$592$	0	0
$593$	12.8643	0.528272	0.264136	−	0.964485i	$-0.414913\pi$
$593$	12.8643	0.528272	0.264136	+	0.964485i	$0.414913\pi$
$594$	0	0
$595$	19.4670	0.798071
$596$	0	0
$597$	−1.29730	−0.0530948
$598$	0	0
$599$	25.5685	1.04470	0.522350	−	0.852731i	$-0.325056\pi$
$599$	25.5685	1.04470	0.522350	+	0.852731i	$0.325056\pi$
$600$	0	0
$601$	6.63763	0.270755	0.135377	−	0.990794i	$-0.456775\pi$
$601$	6.63763	0.270755	0.135377	+	0.990794i	$0.456775\pi$
$602$	0	0
$603$	18.2810	0.744461
$604$	0	0
$605$	−27.7974	−1.13012
$606$	0	0
$607$	22.3226	0.906047	0.453023	−	0.891499i	$-0.350345\pi$
$607$	22.3226	0.906047	0.453023	+	0.891499i	$0.350345\pi$
$608$	0	0
$609$	6.59975	0.267435
$610$	0	0
$611$	0	0
$612$	0	0
$613$	4.23671	0.171119	0.0855596	−	0.996333i	$-0.472732\pi$
$613$	4.23671	0.171119	0.0855596	+	0.996333i	$0.472732\pi$
$614$	0	0
$615$	3.58518	0.144568
$616$	0	0
$617$	14.0193	0.564397	0.282198	−	0.959356i	$-0.408936\pi$
$617$	14.0193	0.564397	0.282198	+	0.959356i	$0.408936\pi$
$618$	0	0
$619$	−23.8469	−0.958487	−0.479244	−	0.877682i	$-0.659089\pi$
$619$	−23.8469	−0.958487	−0.479244	+	0.877682i	$0.659089\pi$
$620$	0	0
$621$	21.4438	0.860510
$622$	0	0
$623$	−11.8964	−0.476619
$624$	0	0
$625$	−31.1402	−1.24561
$626$	0	0
$627$	−3.39090	−0.135420
$628$	0	0
$629$	−43.1809	−1.72174
$630$	0	0
$631$	−7.18624	−0.286080	−0.143040	−	0.989717i	$-0.545688\pi$
$631$	−7.18624	−0.286080	−0.143040	+	0.989717i	$0.545688\pi$
$632$	0	0
$633$	−33.9777	−1.35049
$634$	0	0
$635$	30.5958	1.21416
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.04655	0.199639
$640$	0	0
$641$	8.05886	0.318306	0.159153	−	0.987254i	$-0.449124\pi$
$641$	8.05886	0.318306	0.159153	+	0.987254i	$0.449124\pi$
$642$	0	0
$643$	−28.4148	−1.12057	−0.560286	−	0.828300i	$-0.689309\pi$
$643$	−28.4148	−1.12057	−0.560286	+	0.828300i	$0.689309\pi$
$644$	0	0
$645$	2.03083	0.0799640
$646$	0	0
$647$	−37.6791	−1.48132	−0.740659	−	0.671881i	$-0.765487\pi$
$647$	−37.6791	−1.48132	−0.740659	+	0.671881i	$0.765487\pi$
$648$	0	0
$649$	12.3158	0.483437
$650$	0	0
$651$	−4.90128	−0.192097
$652$	0	0
$653$	37.0486	1.44982	0.724911	−	0.688842i	$-0.241881\pi$
$653$	37.0486	1.44982	0.724911	+	0.688842i	$0.241881\pi$
$654$	0	0
$655$	−18.4341	−0.720282
$656$	0	0
$657$	−4.23269	−0.165133
$658$	0	0
$659$	44.4553	1.73173	0.865867	−	0.500274i	$-0.166767\pi$
$659$	44.4553	1.73173	0.865867	+	0.500274i	$0.166767\pi$
$660$	0	0
$661$	−40.3413	−1.56909	−0.784547	−	0.620069i	$-0.787105\pi$
$661$	−40.3413	−1.56909	−0.784547	+	0.620069i	$0.787105\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.10175	0.275394
$666$	0	0
$667$	−19.5466	−0.756849
$668$	0	0
$669$	−22.9671	−0.887959
$670$	0	0
$671$	14.3025	0.552143
$672$	0	0
$673$	16.8024	0.647684	0.323842	−	0.946111i	$-0.395025\pi$
$673$	16.8024	0.647684	0.323842	+	0.946111i	$0.395025\pi$
$674$	0	0
$675$	15.8469	0.609947
$676$	0	0
$677$	−46.5650	−1.78964	−0.894819	−	0.446429i	$-0.852696\pi$
$677$	−46.5650	−1.78964	−0.894819	+	0.446429i	$0.852696\pi$
$678$	0	0
$679$	8.60355	0.330174
$680$	0	0
$681$	−23.0622	−0.883746
$682$	0	0
$683$	5.79498	0.221739	0.110869	−	0.993835i	$-0.464636\pi$
$683$	5.79498	0.221739	0.110869	+	0.993835i	$0.464636\pi$
$684$	0	0
$685$	−35.9135	−1.37218
$686$	0	0
$687$	−5.45164	−0.207993
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−27.1213	−1.03174	−0.515871	−	0.856666i	$-0.672532\pi$
$691$	−27.1213	−1.03174	−0.515871	+	0.856666i	$0.672532\pi$
$692$	0	0
$693$	−1.37018	−0.0520489
$694$	0	0
$695$	−52.0506	−1.97439
$696$	0	0
$697$	−6.88914	−0.260945
$698$	0	0
$699$	−36.5941	−1.38412
$700$	0	0
$701$	−11.0283	−0.416535	−0.208267	−	0.978072i	$-0.566782\pi$
$701$	−11.0283	−0.416535	−0.208267	+	0.978072i	$0.566782\pi$
$702$	0	0
$703$	−15.7528	−0.594127
$704$	0	0
$705$	26.0318	0.980415
$706$	0	0
$707$	−8.36910	−0.314752
$708$	0	0
$709$	−49.7496	−1.86839	−0.934193	−	0.356768i	$-0.883879\pi$
$709$	−49.7496	−1.86839	−0.934193	+	0.356768i	$0.883879\pi$
$710$	0	0
$711$	−9.72666	−0.364778
$712$	0	0
$713$	14.5162	0.543638
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.5700	−0.432090
$718$	0	0
$719$	−28.5650	−1.06529	−0.532647	−	0.846337i	$-0.678803\pi$
$719$	−28.5650	−1.06529	−0.532647	+	0.846337i	$0.678803\pi$
$720$	0	0
$721$	−0.521442	−0.0194195
$722$	0	0
$723$	−21.9942	−0.817972
$724$	0	0
$725$	−14.4449	−0.536470
$726$	0	0
$727$	−36.5250	−1.35464	−0.677318	−	0.735691i	$-0.736858\pi$
$727$	−36.5250	−1.35464	−0.677318	+	0.735691i	$0.736858\pi$
$728$	0	0
$729$	26.0464	0.964681
$730$	0	0
$731$	−3.90237	−0.144334
$732$	0	0
$733$	18.7444	0.692340	0.346170	−	0.938172i	$-0.387482\pi$
$733$	18.7444	0.692340	0.346170	+	0.938172i	$0.387482\pi$
$734$	0	0
$735$	−3.62016	−0.133532
$736$	0	0
$737$	−14.2345	−0.524336
$738$	0	0
$739$	11.1838	0.411405	0.205702	−	0.978615i	$-0.434052\pi$
$739$	11.1838	0.411405	0.205702	+	0.978615i	$0.434052\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32.7986	1.20326	0.601632	−	0.798774i	$-0.294518\pi$
$743$	32.7986	1.20326	0.601632	+	0.798774i	$0.294518\pi$
$744$	0	0
$745$	27.9195	1.02289
$746$	0	0
$747$	−22.0549	−0.806948
$748$	0	0
$749$	9.43523	0.344756
$750$	0	0
$751$	28.7451	1.04893	0.524463	−	0.851433i	$-0.324266\pi$
$751$	28.7451	1.04893	0.524463	+	0.851433i	$0.324266\pi$
$752$	0	0
$753$	−8.67554	−0.316154
$754$	0	0
$755$	39.3304	1.43138
$756$	0	0
$757$	12.4952	0.454145	0.227072	−	0.973878i	$-0.427085\pi$
$757$	12.4952	0.454145	0.227072	+	0.973878i	$0.427085\pi$
$758$	0	0
$759$	−5.11944	−0.185824
$760$	0	0
$761$	−29.6628	−1.07528	−0.537639	−	0.843175i	$-0.680684\pi$
$761$	−29.6628	−1.07528	−0.537639	+	0.843175i	$0.680684\pi$
$762$	0	0
$763$	13.6860	0.495465
$764$	0	0
$765$	−25.8236	−0.933656
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−40.8488	−1.47305	−0.736523	−	0.676412i	$-0.763534\pi$
$769$	−40.8488	−1.47305	−0.736523	+	0.676412i	$0.763534\pi$
$770$	0	0
$771$	−18.9718	−0.683251
$772$	0	0
$773$	19.4045	0.697932	0.348966	−	0.937135i	$-0.386533\pi$
$773$	19.4045	0.697932	0.348966	+	0.937135i	$0.386533\pi$
$774$	0	0
$775$	10.7275	0.385341
$776$	0	0
$777$	8.03007	0.288077
$778$	0	0
$779$	−2.51322	−0.0900453
$780$	0	0
$781$	−3.92951	−0.140609
$782$	0	0
$783$	−28.5540	−1.02044
$784$	0	0
$785$	−21.7400	−0.775935
$786$	0	0
$787$	10.0913	0.359717	0.179858	−	0.983693i	$-0.442436\pi$
$787$	10.0913	0.359717	0.179858	+	0.983693i	$0.442436\pi$
$788$	0	0
$789$	7.79575	0.277536
$790$	0	0
$791$	−12.2655	−0.436111
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−4.26563	−0.151286
$796$	0	0
$797$	11.2735	0.399329	0.199665	−	0.979864i	$-0.436015\pi$
$797$	11.2735	0.399329	0.199665	+	0.979864i	$0.436015\pi$
$798$	0	0
$799$	−50.0217	−1.76964
$800$	0	0
$801$	15.7810	0.557593
$802$	0	0
$803$	3.29579	0.116306
$804$	0	0
$805$	10.7219	0.377897
$806$	0	0
$807$	1.06788	0.0375912
$808$	0	0
$809$	17.4236	0.612581	0.306290	−	0.951938i	$-0.400912\pi$
$809$	17.4236	0.612581	0.306290	+	0.951938i	$0.400912\pi$
$810$	0	0
$811$	35.9555	1.26257	0.631284	−	0.775552i	$-0.282528\pi$
$811$	35.9555	1.26257	0.631284	+	0.775552i	$0.282528\pi$
$812$	0	0
$813$	−12.7132	−0.445871
$814$	0	0
$815$	20.6902	0.724745
$816$	0	0
$817$	−1.42362	−0.0498061
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.1183	−0.527631	−0.263816	−	0.964573i	$-0.584981\pi$
$821$	−15.1183	−0.527631	−0.263816	+	0.964573i	$0.584981\pi$
$822$	0	0
$823$	−43.2335	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$823$	−43.2335	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$824$	0	0
$825$	−3.78325	−0.131716
$826$	0	0
$827$	50.4513	1.75436	0.877182	−	0.480158i	$-0.159421\pi$
$827$	50.4513	1.75436	0.877182	+	0.480158i	$0.159421\pi$
$828$	0	0
$829$	−1.79727	−0.0624219	−0.0312110	−	0.999513i	$-0.509936\pi$
$829$	−1.79727	−0.0624219	−0.0312110	+	0.999513i	$0.509936\pi$
$830$	0	0
$831$	36.0297	1.24985
$832$	0	0
$833$	6.95635	0.241023
$834$	0	0
$835$	43.9700	1.52165
$836$	0	0
$837$	21.2056	0.732971
$838$	0	0
$839$	−4.72104	−0.162988	−0.0814942	−	0.996674i	$-0.525969\pi$
$839$	−4.72104	−0.162988	−0.0814942	+	0.996674i	$0.525969\pi$
$840$	0	0
$841$	−2.97220	−0.102490
$842$	0	0
$843$	5.22957	0.180116
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−9.93311	−0.341306
$848$	0	0
$849$	−38.4978	−1.32124
$850$	0	0
$851$	−23.7828	−0.815265
$852$	0	0
$853$	48.8428	1.67234	0.836172	−	0.548467i	$-0.184788\pi$
$853$	48.8428	1.67234	0.836172	+	0.548467i	$0.184788\pi$
$854$	0	0
$855$	−9.42069	−0.322181
$856$	0	0
$857$	−14.9964	−0.512267	−0.256134	−	0.966641i	$-0.582449\pi$
$857$	−14.9964	−0.512267	−0.256134	+	0.966641i	$0.582449\pi$
$858$	0	0
$859$	−33.0464	−1.12753	−0.563764	−	0.825936i	$-0.690647\pi$
$859$	−33.0464	−1.12753	−0.563764	+	0.825936i	$0.690647\pi$
$860$	0	0
$861$	1.28113	0.0436607
$862$	0	0
$863$	−47.1122	−1.60372	−0.801859	−	0.597513i	$-0.796156\pi$
$863$	−47.1122	−1.60372	−0.801859	+	0.597513i	$0.796156\pi$
$864$	0	0
$865$	1.16300	0.0395430
$866$	0	0
$867$	−40.6080	−1.37912
$868$	0	0
$869$	7.57367	0.256919
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−11.4129	−0.386267
$874$	0	0
$875$	−6.06884	−0.205164
$876$	0	0
$877$	6.74601	0.227797	0.113898	−	0.993492i	$-0.463666\pi$
$877$	6.74601	0.227797	0.113898	+	0.993492i	$0.463666\pi$
$878$	0	0
$879$	−38.5618	−1.30066
$880$	0	0
$881$	−22.5679	−0.760333	−0.380166	−	0.924918i	$-0.624133\pi$
$881$	−22.5679	−0.760333	−0.380166	+	0.924918i	$0.624133\pi$
$882$	0	0
$883$	19.2628	0.648245	0.324122	−	0.946015i	$-0.394931\pi$
$883$	19.2628	0.648245	0.324122	+	0.946015i	$0.394931\pi$
$884$	0	0
$885$	−43.1647	−1.45097
$886$	0	0
$887$	−0.852658	−0.0286295	−0.0143147	−	0.999898i	$-0.504557\pi$
$887$	−0.852658	−0.0286295	−0.0143147	+	0.999898i	$0.504557\pi$
$888$	0	0
$889$	10.9331	0.366685
$890$	0	0
$891$	−3.36802	−0.112833
$892$	0	0
$893$	−18.2483	−0.610658
$894$	0	0
$895$	−10.9361	−0.365555
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−19.3295	−0.644674
$900$	0	0
$901$	8.19667	0.273071
$902$	0	0
$903$	0.725697	0.0241497
$904$	0	0
$905$	−13.4088	−0.445725
$906$	0	0
$907$	−46.7594	−1.55262	−0.776310	−	0.630351i	$-0.782911\pi$
$907$	−46.7594	−1.55262	−0.776310	+	0.630351i	$0.782911\pi$
$908$	0	0
$909$	11.1019	0.368226
$910$	0	0
$911$	−33.7415	−1.11791	−0.558954	−	0.829199i	$-0.688797\pi$
$911$	−33.7415	−1.11791	−0.558954	+	0.829199i	$0.688797\pi$
$912$	0	0
$913$	17.1731	0.568347
$914$	0	0
$915$	−50.1279	−1.65718
$916$	0	0
$917$	−6.58725	−0.217530
$918$	0	0
$919$	−23.4340	−0.773018	−0.386509	−	0.922286i	$-0.626319\pi$
$919$	−23.4340	−0.773018	−0.386509	+	0.922286i	$0.626319\pi$
$920$	0	0
$921$	24.0521	0.792545
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−17.5754	−0.577876
$926$	0	0
$927$	0.691710	0.0227187
$928$	0	0
$929$	−10.9826	−0.360328	−0.180164	−	0.983637i	$-0.557663\pi$
$929$	−10.9826	−0.360328	−0.180164	+	0.983637i	$0.557663\pi$
$930$	0	0
$931$	2.53774	0.0831709
$932$	0	0
$933$	20.5841	0.673893
$934$	0	0
$935$	20.1076	0.657589
$936$	0	0
$937$	47.5028	1.55185	0.775924	−	0.630826i	$-0.217284\pi$
$937$	47.5028	1.55185	0.775924	+	0.630826i	$0.217284\pi$
$938$	0	0
$939$	2.32302	0.0758090
$940$	0	0
$941$	8.08915	0.263699	0.131849	−	0.991270i	$-0.457908\pi$
$941$	8.08915	0.263699	0.131849	+	0.991270i	$0.457908\pi$
$942$	0	0
$943$	−3.79434	−0.123561
$944$	0	0
$945$	15.6627	0.509508
$946$	0	0
$947$	18.7265	0.608528	0.304264	−	0.952588i	$-0.401589\pi$
$947$	18.7265	0.608528	0.304264	+	0.952588i	$0.401589\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	20.7382	0.672482
$952$	0	0
$953$	−47.2451	−1.53042	−0.765209	−	0.643781i	$-0.777365\pi$
$953$	−47.2451	−1.53042	−0.765209	+	0.643781i	$0.777365\pi$
$954$	0	0
$955$	−24.1480	−0.781412
$956$	0	0
$957$	6.81692	0.220360
$958$	0	0
$959$	−12.8333	−0.414409
$960$	0	0
$961$	−16.6450	−0.536936
$962$	0	0
$963$	−12.5161	−0.403326
$964$	0	0
$965$	−68.4320	−2.20290
$966$	0	0
$967$	−39.7889	−1.27952	−0.639762	−	0.768573i	$-0.720967\pi$
$967$	−39.7889	−1.27952	−0.639762	+	0.768573i	$0.720967\pi$
$968$	0	0
$969$	−22.8369	−0.733626
$970$	0	0
$971$	10.0133	0.321341	0.160670	−	0.987008i	$-0.448634\pi$
$971$	10.0133	0.321341	0.160670	+	0.987008i	$0.448634\pi$
$972$	0	0
$973$	−18.5998	−0.596281
$974$	0	0
$975$	0	0
$976$	0	0
$977$	58.9087	1.88466	0.942328	−	0.334691i	$-0.108632\pi$
$977$	58.9087	1.88466	0.942328	+	0.334691i	$0.108632\pi$
$978$	0	0
$979$	−12.2879	−0.392722
$980$	0	0
$981$	−18.1549	−0.579640
$982$	0	0
$983$	−5.15605	−0.164452	−0.0822262	−	0.996614i	$-0.526203\pi$
$983$	−5.15605	−0.164452	−0.0822262	+	0.996614i	$0.526203\pi$
$984$	0	0
$985$	−19.9846	−0.636761
$986$	0	0
$987$	9.30221	0.296092
$988$	0	0
$989$	−2.14931	−0.0683442
$990$	0	0
$991$	−62.6905	−1.99143	−0.995715	−	0.0924728i	$-0.970523\pi$
$991$	−62.6905	−1.99143	−0.995715	+	0.0924728i	$0.970523\pi$
$992$	0	0
$993$	−8.81982	−0.279889
$994$	0	0
$995$	2.80640	0.0889687
$996$	0	0
$997$	−14.1377	−0.447745	−0.223872	−	0.974618i	$-0.571870\pi$
$997$	−14.1377	−0.447745	−0.223872	+	0.974618i	$0.571870\pi$
$998$	0	0
$999$	−34.7423	−1.09920

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.a.p.1.2		4
13.5	odd	4		364.2.g.a.337.3	✓	8
13.8	odd	4		364.2.g.a.337.4	yes	8
13.12	even	2		4732.2.a.o.1.2		4
39.5	even	4		3276.2.e.f.2521.8		8
39.8	even	4		3276.2.e.f.2521.1		8
52.31	even	4		1456.2.k.d.337.5		8
52.47	even	4		1456.2.k.d.337.6		8
91.5	even	12		2548.2.y.e.753.4		16
91.18	odd	12		2548.2.y.f.961.6		16
91.31	even	12		2548.2.y.e.961.3		16
91.34	even	4		2548.2.g.g.2157.5		8
91.44	odd	12		2548.2.y.f.753.5		16
91.47	even	12		2548.2.y.e.753.3		16
91.60	odd	12		2548.2.y.f.961.5		16
91.73	even	12		2548.2.y.e.961.4		16
91.83	even	4		2548.2.g.g.2157.6		8
91.86	odd	12		2548.2.y.f.753.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.g.a.337.3	✓	8	13.5	odd	4
364.2.g.a.337.4	yes	8	13.8	odd	4
1456.2.k.d.337.5		8	52.31	even	4
1456.2.k.d.337.6		8	52.47	even	4
2548.2.g.g.2157.5		8	91.34	even	4
2548.2.g.g.2157.6		8	91.83	even	4
2548.2.y.e.753.3		16	91.47	even	12
2548.2.y.e.753.4		16	91.5	even	12
2548.2.y.e.961.3		16	91.31	even	12
2548.2.y.e.961.4		16	91.73	even	12
2548.2.y.f.753.5		16	91.44	odd	12
2548.2.y.f.753.6		16	91.86	odd	12
2548.2.y.f.961.5		16	91.60	odd	12
2548.2.y.f.961.6		16	91.18	odd	12
3276.2.e.f.2521.1		8	39.8	even	4
3276.2.e.f.2521.8		8	39.5	even	4
4732.2.a.o.1.2		4	13.12	even	2
4732.2.a.p.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4732.a (trivial)

\(f(q)\)	\(=\)	\(q-1.29363 q^{3} +2.79846 q^{5} +1.00000 q^{7} -1.32653 q^{9} +O(q^{10})\)	\(q-1.29363 q^{3} +2.79846 q^{5} +1.00000 q^{7} -1.32653 q^{9} +1.03291 q^{11} -3.62016 q^{15} +6.95635 q^{17} +2.53774 q^{19} -1.29363 q^{21} +3.83136 q^{23} +2.83136 q^{25} +5.59691 q^{27} -5.10175 q^{29} +3.78880 q^{31} -1.33619 q^{33} +2.79846 q^{35} -6.20741 q^{37} -0.990338 q^{41} -0.560979 q^{43} -3.71224 q^{45} -7.19080 q^{47} +1.00000 q^{49} -8.99892 q^{51} +1.17830 q^{53} +2.89054 q^{55} -3.28288 q^{57} +11.9234 q^{59} +13.8469 q^{61} -1.32653 q^{63} -13.7811 q^{67} -4.95635 q^{69} -3.80432 q^{71} +3.19080 q^{73} -3.66272 q^{75} +1.03291 q^{77} +7.33240 q^{79} -3.26072 q^{81} +16.6260 q^{83} +19.4670 q^{85} +6.59975 q^{87} -11.8964 q^{89} -4.90128 q^{93} +7.10175 q^{95} +8.60355 q^{97} -1.37018 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 3 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q + 3 q^{5} + 4 q^{7} + 4 q^{9} - 2 q^{17} + 3 q^{19} + 3 q^{23} - q^{25} + 6 q^{27} - q^{29} + 13 q^{31} + 10 q^{33} + 3 q^{35} - 10 q^{41} + 3 q^{43} + 13 q^{45} - 3 q^{47} + 4 q^{49} + 4 q^{51} + 11 q^{53} - 10 q^{55} + 26 q^{57} + 22 q^{59} + 4 q^{61} + 4 q^{63} - 12 q^{67} + 10 q^{69} + 26 q^{71} - 13 q^{73} + 10 q^{75} - 13 q^{79} - 12 q^{81} + 19 q^{83} + 12 q^{85} - 16 q^{87} + 7 q^{89} - 32 q^{93} + 9 q^{95} + 19 q^{97} - 26 q^{99}+O(q^{100}) \) 4 * q + 3 * q^5 + 4 * q^7 + 4 * q^9 - 2 * q^17 + 3 * q^19 + 3 * q^23 - q^25 + 6 * q^27 - q^29 + 13 * q^31 + 10 * q^33 + 3 * q^35 - 10 * q^41 + 3 * q^43 + 13 * q^45 - 3 * q^47 + 4 * q^49 + 4 * q^51 + 11 * q^53 - 10 * q^55 + 26 * q^57 + 22 * q^59 + 4 * q^61 + 4 * q^63 - 12 * q^67 + 10 * q^69 + 26 * q^71 - 13 * q^73 + 10 * q^75 - 13 * q^79 - 12 * q^81 + 19 * q^83 + 12 * q^85 - 16 * q^87 + 7 * q^89 - 32 * q^93 + 9 * q^95 + 19 * q^97 - 26 * q^99

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